Unit 3: Probability
Coincidences
(Expected to occur, likelihoods dictated by the laws of probability.) n=8 r=3 Step One: 8!/(8-3)! = 8!/5! Step Two: 8 x 7 x 6 x 5!/5! Step Three: 5! cancel each other out so you have 8 x 7 x 6 Answer: 336
what is the probability of drawing 2 aces from a deck of cards
4/52 x 3/51 =12/2652 =1/221
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
6/12 x 5/11 x 4/10 = 120/1320 = 0.091
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 4/52 = 1/13 ( one four in each of the 4 suits)
Rolling a single six-sided die and getting a 1 or a 6
The probability of rolling a single six-sided die and getting a 1 or 6 is: 1/6 + 1/6 = 2/6 = 1/3
overlapping events
if they can occur together, like the outcome of picking a queen or a club
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.
Suppose you roll a die 6 times. What is the probability of getting at least one odd number?
(not A) = 3/6 = 1/2 1-P (not A)^n (1/2) ^6 = 1/64 1- 1/64 = 63/64
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-P (not A)^n 1 - (1/2) ^ 9 = 511/512
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
1-P (not A)^n P (at least one wins) = P(9/10) = 1 - (9/10)^10 =0.651
Use the "at least once" rule to find the probability of getting at least one 6 in four rolls of a single fair die.
1-P (not A)^n P(6) = 1/6 P(not 6) = 1 - P(6) = 1 - 1/6 = 5/6 1 - (5/6) ^ 4 = 0.518
Example of At Least One. Winning the grand price in a local drawing is 1 out of 30. Two people buy tickets. What is the probability that at least one of them will win the grand prize?
1-P (not A)^n Step 1: Find the probability of not winning. 29/30 or .96. So, (not A) = (.967) Step 2: n = number of tickets bought or n=2 Step 3: 1 - (.967)^2 Answer: .065 or 6.5%
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
Sharing a birthday with another person when you both have birthdays in October.
1/31 (31 days in October)
suppose you roll a single die, what is the probability of rolling either a 2 or a 3
1/6 + 1/6 = 2/6 =1/3
what is the probability of rolling three 4's in a row with a single die?
1/6 x 1/6 x 1/6 = 1/216
Determine the probability of the given complementary event. What is the probability of randomly selecting a month of the year and not getting February?
11/12
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
An experiment consists of drawing 1 card from a standard 52-card deck. What is the probability of drawing a club?
13/52 = 1/4 (13 of each suit in a deck of cards/52 total cards in a deck)
Use the theoretical method to determine the probability of the following event. A randomly selected person has a birthday in October.
31/365 (31 days in October/365 days in a year)
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 1-P (not A)^n P(not A) = probability of drawing a card that is not a 10 =48/52 = 12/13 1 - (12/13)^4 = 0.2740
Multiplication Principle
If there are "a" ways to complete a task and "b" wasy to complete a second task and no outcomes from the 1st effect the outcome of the second. a x b = Outcomes
what is the probability that in a standard shuffled deck of cards you will draw a 5 or a spade
P(5 or spade) - Three events: Draw a 5, Draw a Spade or Draw the 5 of Spades = P(5) + P(spade) - P(5 and spade) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13
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)
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
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
Law of Large Numbers
The larger the number of trials, the closer proportion should be to the probability.
At Least Once Probability
The probability of the event never occurring and the probability of the event occurring at least once will equal one, or a 100% chance.
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 occurrence 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
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
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) .2 + .4 + .6 + .8 + 1 = 3
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.
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: 6/8 = 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.
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.
Does this statement make sense: 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.
non overlapping events
They cannot occur together, like the outcome of a coin toss (heads or tails)
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.
Example of a situation that would we be interested in an either/or probability?
We want to know whether a person selected at random is a Democrat or a man.
Example of an overlapping event
We want to know whether a person selected at random is a Democrat or a man.
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 EV(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
Is the following situations the events non-overlapping? We roll a die, hoping for a 2 or a 5.
Yes it is non-overlapping. You can't do both at the same time. We roll a die, hoping for a 2 or a 5.
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?
You could roll a 5 or a 6 so, 2/6 = 1/3
Example of a Dependent Probability
You have a deck of cards. What is the probability of drawing a Queen in three draws and you do not return the card to the deck. 4/52 x 3/51 x 2/50 = 24/132600 = 1/5525
Drawing three jacks in a row from a standard deck of cards when the drawn card is not returned to the deck each time
dependent and probability ( without replacement ) P(A) =4/52 = 1/13 P(B given A) = 3/51 = 1/17 P(C given A and B) = 2/50 = 1/25 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
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
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)
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.
Repetition
n x n x ...x n = n^r r= Selections made n=groups (Different arrangements are possible and can be used over and over)
Permutations
n! (single group of items, can not be used more than once and order does matter)
Combinations
n!/(n-r)! x r! r = Selections made n = Items (not looking at order, single group of items, not selected more than once)
Drawing either a spade or a club from a regular deck of cards
non overlapping 13/52 + 13/52 = 26/52 = 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
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
probability distribution
represents the probabilities of all possible events of interest. The sum of all the probabilities in a probability distribution must be 1.