7C the law of large numbers, 7B combining probabilities, 7A fundamentals of probability
What is the probability of getting a parking ticket on campus on at least one out of 6 days without a parking pass if the chance of not getting a ticket on a particular day when a parking pass isn't present is 0.15?
(0.15)^6 = 0.000 1 - 0.000 = 100%
Purchasing 5 winning lottery tickets in a row when each ticket has a 1 in 5 chance of being a winner
(1/5)^ 5 = 1/3125
Use the "at least once" rule to find the probabilities of the following event. Getting at least one head when tossing nine fair coins
1 - (1/2) ^ 9 = 511/512
Getting rain at least once in 4 days if the probability of rain on each single day is 0.5
1 - (1/2) ^4 = 0.938
At least how many times do you have to roll a fair die to be sure that the probability of rolling at least one 2 is greater than 8 in 10 (0.80)?
1 - (5/6) ^ 9 9 rolls ( test all exponents until the decimal is greater than 0.8)
Suppose you roll a die 4 times. What is the probability of getting at least one six?
1 - 1/6 = 5/6 (5/6)^4 = 625/1296 1 - 625/1296 = 671/1296
suppose you have 5 cards, 1,2,3,4,5. the number on each card represents the number of points you receive if you draw that card. You draw one card at random, what is the expected value?
1(1/5) + 2(1/5) + 3(1/5) + 4(1/5) + 5(1/5) = 3
Determine the probability of the given complementary event. What is the probability of randomly selecting a month of the year and not getting February?
1- 1/12 = 11/12
what is the probability of drawing at least 1 ace when you draw a card from a standard deck 6 times, replacing the card each time
1-P (no A)^6 = 1 - (12/13)^6 = 1 - 0.619 = 0.381
Find the odds for and the odds against the event rolling a fair die and getting a 1, 3, or a 2
1/2 of getting a 1,2, or 3. 1/2 probability of not getting a 1,2 or 3. the odds for the event are 1 to 1 odds against the event are 1 to 1
Spinning two winners in a row with a wheel of fortune on which the winner is one of 29 equally likely outcomes.
1/29 x 1/29 = 1/841
An experiment consists of drawing 1 card from a standard 52-card deck. What is the probability of drawing a club?
1/4
Randomly selecting a four-person committee consisting entirely of Americans from a pool of 12 Americans and 16 Canadians.
12/28 x 11/27 x 10/26 x 9/25 = 0.0242
Determine the probability of the given opposite event. What is the probability of rolling a fair die and not getting an outcome less than 5?
2/6 = 1/3
The local weather forecast has been accurate for 20 of the past 37 days. what is the relative frequency probability that the forecast for tomorrow will be accurate?
20/37 = 0.541
Use the theoretical method to determine the probability of the following event. Sharing a birthday with another person when you both have birthdays in February (not a leap year)
28/961 = 1/28
Suppose you roll a die 6 times. What is the probability of getting at least one odd number?
3/6 = 1/2 (1/2) ^6 = 1/64 1- 1/64 = 63/64
Use the theoretical method to determine the probability of the following event. A randomly selected person has a birthday in April.
30/365 = 6/73
Use the theoretical method to determine the probability of the following event. A randomly selected person has a birthday in October.
31/365
Use the theoretical method to determine the probability of the following event. Sharing a birthday with another person when you both have birthdays in October.
31/961 = 1/31
Drawing at least one ten when you draw a card from a standard deck 4 times (replacing the card each time you draw)
4 tens in 52 cards 48 cards aren't 10s P(not A) = probability of drawing a card that is not a 10 = 12/13 1 - (12/13)^4 = 0.2740
Drawing three jacks in a row from a standard deck of cards when the drawn card is not returned to the deck each time
4/52 x 3/51 x 2/50 = 1/5225
Explain the meaning of the law of large numbers. Does this law say anything about what will happen in a single observation or experiment? Why or why not?
As the experiment is done more and more times, the proportion of times that a certain outcome occurs should get closer to the theoretical probability that that outcome would occur. This law does not say anything about what will happen in a single observation or experiment. Large numbers of events may show some pattern, but the individual events are unpredictable.
Use the theoretical method to determine the probability of the following outcome and event. State any assumptions made. Tossing two coins and getting either one head or two heads
Assuming that each coin is fair and is equally likely to land heads or tails, the probability is 3/4.
What is an expected value and how is it computed? Should we always expect to get the expected value? Why or why not?
Expected value is the estimated gain or loss of partaking in an event many times. EV = (event 1 value) x (event 1 probability) + (event 2 value) x (event 2 probability) We should not always expect to get the expected value because expected value is calculated with the assumption that the law of large numbers will come into play.
suppose you roll a single die, what is the probability of rolling either a 2 or a 3
P (2 or 3) = P (2) + P(3) = 1/6 + 1/6 = 2/6 =1/3
a 3 person jury must be selected at random from a pool that has 6 men and 6 women. what is the probability of selecting an all male jury
P (all male)= P (1st is male, 2nd is male, 3rd is male) = 6/12 x 5/11 x 4/10 = 120/1320 = 0.091
use the at least once rule to find the probability of at least one head when you toss three coins
P (at least one head) = 1 - P (no head) = 1 - (1/2)(1/2)(1/2) = 7/8
you purchase 10 lottery tickets with the probability of winning 1 in 10. what is the probability that you will have at least 1 winning ticket among the ten
P (at least one wins) = 1 - P (none win) = 1 - (9/10)^10 =0.651
Purchasing at least one winning lottery ticket out of 10 tickets when the probability of winning is 0.04 on a single ticket
P (no A in one trial) = 1 - 0.04 = 0.96 1 - 0.96^10 = 0.3352
what is the probability of rolling three 4's in a row with a single die?
P(4 on first roll , 4 on second roll, 4 on third roll) = P(4 on 1st roll) x P(4 on 2nd roll)x P(4 on 3rd roll) = 1/6 x 1/6 x 1/6 = 1/216
what is the probability that in a standard shuffled deck of cards you will draw a 5 or a spade
P(5 or spade) = P(5) + P(spade) - P(5 and spade) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13
Use the "at least once" rule to find the probability of getting at least one 6 in four rolls of a single fair die.
P(6) = 1/6 P(not 6) = 1 - P(6) = 1 - 1/6 = 5/6 1 - (5/6) ^ 4 = 0.518
If A and B are non-overlapping events, then
P(A or B) = P(A) + P(B)
If A and B are overlapping events, then
P(A or B) = P(A) + P(B) - P(A and B)
Getting a green light at a busy intersection at least once in six times through the intersection, given that the light in your direction is green 4/10 of the time
P(A) = 0.4 P(not A) = 1 - 0.4 = 0.6 n = 6 1 - (0.6)^6 = 0.953
what is the probability of drawing 2 aces from a deck of cards
P(A1) x P(A2/A1) = 1/13 x 3/51
a roulette wheel has 38 numbers: 18 black, 18 red, and the numbers 0 and 00 in green. assume all possible outcomes (the 38 #'s) have equal probability what is the probability of getting a red number on any spin if patrons spin the wheel 100,000 times, about how many times will a red number be the outcome
P(red)= number of ways red can occur ---------------------------------- total number of outcomes = 18/38 = 0.474 law of large numbers tells us that as the game is played more, the proportion of times that the wheel shows a red number should get closer to 0.474. in 100,000 tries, the wheel should come up red close to 47.6% of the time or about 47,400
house edge
The "house" sets the odds on casino games to ensure a profit over a period of time. the expected value to the house of each individual bet
Rolling two 6s followed by one 3 on three tosses of a fair die.
The individual events are independent. The probability of the combined event is (1/6) x (1/6) x (1/6) = 1/216
Use the theoretical method to determine the probability of the given outcome or event. Assume that the die is fair. Rolling a single six-sided die and getting a 1 or a 6
The probability rolling a single six-sided die and getting a 1 or 6 is 1/3
If you toss a coin four times, it's much more likely to land in the order HTHT than HHHH. (H stands for heads and T for tails.)
The statement does not make sense because each outcome is equally likely since the probability of any single particular outcome is 1/2, so each set of outcomes have the same probability of (1/2) ^4 = 1/16
I haven't won in my last 25 pulls on the slot machine, so I must be having a bad day and I'm sure to lose if I play again.
The statement does not make sense because the results of repeated trials do not depend on results of earlier trials.
The probability that Jonas will win the race is 0.6 and the probability that he will not win is 0.5.
The statement does not make sense because the sum of the probabilities of Jonas winning and not winning the race must equal to 1.
The probability of drawing an ace or a spade from a deck of cards is the same as the probability of drawing the ace of spades.
The statement does not make sense because there is one card that is the ace of spades but more than one card that is either an ace or a spade.
Because either there is life on Mars or there is not, the probability of life on Mars is 0.5.
The statement does not make sense. Although there are two possible outcomes, it is not reasonable to assume that both outcomes are equally likely.
When I toss four coins, there are six different outcomes that all represent the event of four heads.
The statement does not make sense. There is only one way the event can occur.
The expected value to me of each raffle ticket I purchased is - $0.85.
The statement makes sense because a negative expected value implies that, averaged over many tickets, you should expect to lose $0.85 for each raffle ticket that you buy.
The probability of getting heads and tails when you toss a coin is 0, but the probability of getting heads or tails is 1.
The statement makes sense because heads and tails are the only possible outcomes and it is impossible to get both heads and tails on a single coin toss.
State which method (theoretical, relative frequency, or subjective) should be used to answer the question below. What is the probability of being dealt a three-of-a-kind of fives in a five-card hand?
The theoretical method should be used because the best approach to finding the probability is counting all of the possible hands with a three-of-a-kind of fives.
Explain why the probability is the same for any particular set of ten coin toss outcomes. How does this idea affect our thinking about streaks?
The total number of outcomes for ten coins is 2 x 2 x 2 x 2 x 2 x 2 x 2 2 x 2 x 2 = 1024, so every individual outcome has the same probability of 1 divided by 1024. A streak of all heads would not seem surprising since a streak of all heads is just as likely as a streak of all tails, or as likely as any other combination of outcomes.
My chance of getting a 5 on the roll of one die is 1/6, so my chance of getting at least one 5 when I roll three dice is 3/6.
This does not make sense because the real probability would be 1 - (5/6) ^3 , which is not equal to 3 divided by 6.
In which of the following situations are the events non-overlapping? Select all that apply.We roll a die, hoping for a 2 or a 5.
We roll a die, hoping for a 2 or a 5.
In which of the following situations are the events overlapping? Select all that apply.
We want to know whether a person selected at random is a Democrat or a man.
In which of the following situations would we be interested in an either/or probability? Select all that apply.
We want to know whether a person selected at random is a Democrat or a man. We roll a die, hoping for a 2 or a 5.
a company makes electronics. one out of every 50 gadgets is faulty but the company doesn't know until a customer complains. suppose the company makes a $3 profit on the sale of a gadget but suffers an $80 loss for every faulty gadget because they have to repair. can the company expect a profit in the long term?
X= profit E(X)= $3(49/50) - $80(1/50) = 147/50 - 80/50 = 67/50 = 1.34 since the expected value is positive the company can expect to make a profit
you toss a coin 300 times and get 42 tails. Complete parts (a)dash(c) below. (a) Determine the relative frequency probability. (b) Determine the expected frequency of the event. (c) Do you have reason to suspect the coin is unfair? Explain.
a. 0.14 b. 0.5 c. yes because the expected frequency is nowhere near the relative frequency
suppose we describe the weather as either hot (Upper H) or frigid (Upper F). Answer parts (a) and (b) below. a. Using the letters Upper H and Upper F, list all the possible outcomes for the weather on two consecutive days. b. If we are only interested in the number of Upper H days, what are the possible events for two consecutive days.
a. HH, HF, FH, FF b. 0, 1, 2
Randomly drawing and immediately eating two red pieces of candy in a row from a bag that contains 10 red pieces of candy out of 43 pieces of candy total.
dependent 10/43 x 9/42 = 0.050
Being dealt three jacks off the top of a standard deck of well-shuffled cards.
dependent and probability ( without replacement ) P(A and B and C) = P(A) x P(B given A) x P(C given A and B) 1/13 x 1/17 x 1/25 = 1/5525
Drawing three red cards in a row from a standard deck of cards when the drawn card is not returned to the deck each time
dependent, 2/17 ( 0.1176)
overlapping events
if they can occur together, like the outcome of picking a queen or a club the probability that either A or B occurs: P (A or B) = P(A) + P(B) - P(A and B)
explain how to make a table of a probability distribution
list all possible outcomes, identify the outcomes that represent the same event, and then find the probability of each event.
Drawing either a spade or a club from a regular deck of cards
non overlapping 13/52 + 13/52 = 1/2
Drawing either a black eight or a red three on one draw from a regular deck of cards
non overlapping, probability is 2/52 + 2/52 = 1/26 + 1/26 = 1/13
Use the theoretical method to determine the probability of the outcome or event given below. The next president of the United States was born on Tuesday
number of ways being born on a tuesday can occur --------------------------------------------------------- total number of outcomes = 1/7
Use the definitions given in the text to find both the odds for and the odds against the following event. Flipping 4 fair coins and getting exactly 1 head.
odds of one head: 1-3 odds against 1 head: 3-1
probability distribution
represents the probabilities of all possible events of interest. The sum of all the probabilities in a probability distribution must be 1.
At least once rule (independent events)
suppose probability of an event A occurring in one trial is P(A). if all trials are independent, the probability that event A occurs at least once in n trials is: P (at least 1 event) = 1 - P (no event A)
gambler's fallacy
the mistaken belief that a streak of bad luck makes a person "due" for a streak of good luck
independent events
the occurence of one event does not affect the probability of the other event occurring.
dependent events
the outcome of one event affects the outcome of another event
An experiment consists of drawing 1 card from a standard 52-card deck. What is the probability of drawing a 4?
the probability of drawing a 4 is 1/13 ( one four in each 13 card suit )
non overlapping events
they cannot occur together, like the outcome of a coin toss (heads or tails) the probability for non-overlapping events A and B, P(A or B) = P(A) + P(B)
Suppose you toss a fair coin 10,000 times. Should you expect to get exactly 5000 heads? Why or why not? What does the law of large numbers tell you about the results you are likely to get?
you shouldn't expect to get exactly 5000 heads, because you cannot predict precisely how many heads will occur. The proportion of heads should approach 0.5 as the number of tosses increases.
