Preparation W08 Math 108

Give a thoughtful example for each of the three types of probabilities: (empirical, theoretical, subjective).

1. Consider 3 pieces of buttered bread. How likely is it for the event of two buttered pieces to land butter side up and one piece butter side down? BBB BBD BDB BDD DBB DBD DDB DDD - Of the 8 possible outcomes, 3 have two butter up and one down. The probability that in a toss two will land up and one will land down is 3/8. This is theoretical. 2. An airplane has flown over the SL valley 5 times in 3 hours. What is the relative frequency that a plane will fly over in the next hour? Number of times A occurred/total number of observations. 5/3 or 1.666666667. This is relative or empirical. 3. I have a 90% chance of getting the job I applied for. This is subjective.

Using the Multiplication Principle and given that random digits 0 through 9 are allowed for each of the five numbers required, what is the probability that this vehicle license plate from Madison County, Idaho would be assigned the number 56,789?

10x10x10x10x10 = 100,000 so 1:100,000. Since each of the five digits has 10 possibilities, we know that there are 105 = 100,000 possible license plates. Therefore this particular one (despite the fact that it is a sequence) has a 1 out 100,000 probability

If you have 14 shirts and 5 pairs of pants, how many different outfits could you wear? Assume that all of the possibilities look good enough together that you actually would wear them.

14 x 5 = 70 possible outfits

In 2010, there were 40 million people over 65 years of age out of a U.S. population of 310 million. By 2050, it is estimated that there will be 82 million people over 65 years of age out of a U.S. population of 439 million. Would your chances of meeting a person over 65 at random be greater in 2010 or 2050? Explain your answer with some calculated probabilities. (See Problem #64 of 7A in your textbook.) Explain whether this makes sense or not and why:

2010: 40m/310m = .129032258 2050: 82m/439 = .186788155 You are more likely to meet someone over age 65 in 2050 as there are likely more people that age at that time. n 2010 we see that 40/310 = 12.9% of the population will be over 65 years, but in 2050 we see that 82/439 = 18.7%. Therefore, your chances of meeting a person over 65 at random are higher in 2050

You decide to order a new tennis racquet online. The company you are going to buy from offers 3 different sizes of racquet in 8 different colors with 3 different grip styles. Considering the size, color, and grip, how many different tennis racquets are available?

3x8x3 = 72 possible combos

You have 16 pairs of socks in your drawer, of which 4 are black, 5 are brown and 7 are white. If you have to reach into the drawer in the dark and pick a pair of socks, what is the probability that you won't get a black pair?

4 in 16 chance of getting a black pair 12 in 16 of not = .75

When you roll two fair dice, what is the probability of getting a total of 9?

6x6 = 36 possible places for 9 to appear The sum of 9 appears 4 times 4/36 = .111111111 The sum of two dice: 1 2 3 4 5 6 +------------------- 1 | 2 3 4 5 6 7 2 | 3 4 5 6 7 8 3 | 4 5 6 7 8 9 4 | 5 6 7 8 9 10 5 | 6 7 8 9 10 11 6 | 7 8 9 10 11 12

You are getting ready to take the final exam in your math class. So far on all the tests and quizzes, you have passed 85% of them. What are the odds in favor of your passing the final exam?

85% means 15% you did not pass 85/15 reduces to 17:3

A box of candy contains five dark chocolates and five white chocolates. If you pick randomly and eat each candy after choosing it, what is the probability of choosing three dark chocolates in a row? A) 5/10 x 4/9 x 3/8 B) 5/10 x 5/10 x 5/10 C) 5/10 x 4/10 x 3/10

A) 5/10 x 4/9 x 3/8

On a roll of two dice, Serena bets that the sum will be 5, and Mackenzie bets that the sum will be 8. Who has a higher probability of winning? (Hint: See Table 7.3 in your text) A) Mackenzie B) Both have an equal probability of winning. C) Serena

A) Mackenzie

When you toss one coin, the probability that you'll get heads is 1/2. Assuming the coin is fair, this means that A) if you toss 1000 coins, there's no way to predict the precise number of heads you'll get, though it will probably be close to 500. B) if you toss two coins, you'll get 1 head and 1 tail. C) if you toss 100 coins, you'll get 50 heads and 50 tails.

A) if you toss 1000 coins, there's no way to predict the precise number of heads you'll get, though it will probably be close to 500.

The rule P(A and B) = P(A) X P(B) holds A) only if an outcome of A on one trial does not affect the probability of an outcome of B on the next. B) only if it is possible for both A and B to occur together (simultaneously). C) in all cases.

A) only if an outcome of A on one trial does not affect the probability of an outcome of B on the next.

A covered jar contains 3 green marbles, 5 blue marbles, and 9 white marbles and you can't see these contents. What is the probability that, if you randomized the marbles by shaking the jar and pulled out two marbles in a row (without replacement), you would pull either one blue and then one green marble or else pull one green and then one blue marble? (Round to four decimal places.)

AND P(A) = P(B) x P(G) = 5/17 x 3/16 = 15/272 or .0551 P(B) = P(G) x P(B) = 3/17 x 5/16 = 15/272 or .0551 Same probability. OR P(A or B) = P(A) + P(B) = .0551470 + .0551470 = .1103 1 blue / 1 green = 5/17*3/16 = 0.0551 and 1 green / 1 blue = 3/17*5/16 = 0.0551. The two probabilities are then added to equal 0.1103 or about an 11% chance.

Assume that electrical engineers working at Boeing in Seattle, Washington, have calculated that the probability of a particular electronic component in a Boeing 787 Dreamliner has only a 1 in 10,000 chance of going defective in the middle of a flight. Boeing has decided to place a second component next to it to use as a back-up in case of failure. What is the probability that both components would fail during a random flight?

AND; Independent P(A and B) = P(A) x P(B) = 1/10,000 x 1/10,000 = 1/100,000,000 1 in 100 million failure of the components would be an independent event, we would use the AND rule and multiply 1/10,000 by 1/10,000 to get 1/100,000,000 or one in 100 million for the probability.

Suppose the probability of winning a game is X. What is the probability of not winning? A) Y Correct Response B) 1 - X C) P(X)

B) 1 - X 100% - X

You roll two dice twice. Based on the probabilities shown in Table 7.3 (in the text), what is the probability that you'll get a sum of 3 on the first roll and a sum of 4 on the second roll? A) 2/36 + 3/36 B) 2/36 x 3/36 C) (2/36 x 3/36)2

B) 2/36 x 3/36

The events of being born on a Wednesday and being born in July are A) independent. B) overlapping. C) mutually exclusive.

B) overlapping.

During the course of the basketball season, Shawna made 67 out of 100 free throws. When we say that her probability of making a free throw during the playoffs is 0.67, what type of probability are we stating? A) A theoretical probability. B) relative frequency probability (sometimes called empirical probability) C) A subjective probability.

B) relative frequency probability (sometimes called empirical probability)

During the course of the basketball season, Shawna made 67 out of 100 free throws. When we say that her probability of making a free throw during the playoffs is 0.67, what type of probability are we stating? A) A theoretical probability. B) relative frequency probability (sometimes called empirical probability) C) A subjective probability.

B) relative frequency probability (sometimes called empirical probability)

One in 100 tennis balls produced at a factory is defective. If you randomly select five tennis balls, what is the probability that at least one will be defective? A) 5 x 0.015 B) 0.995 C) 1 - 0.995

C) 1 - 0.995

Suppose a women's college basketball player is fouled in a one-and-one situation (meaning she is awarded two free throws only if she makes her first one). Assume that her season average for free throws is 70%. We know that only three following scenarios are possible: A) she makes both, B) she makes the first and then misses the second, or C) she misses the first. Place these three scenarios (A, B, and C) into the order of most likely to happen down to the least likely to happen. (Hint: use a probability tree to help you find the answer. See the video clip below if you need some help calculating the probabilities of the three different scenarios by looking at a slightly different example.)

Event A: 70% x 70% = .49 or 49% Event C: 30% = .30 or 30% Event B: 70% x 30% = .21 or 21% Total = 100%

"The probability of getting a HEAD and a TAIL when you toss a coin is 0, but the probability of getting a HEAD or a TAIL is 1." What is the difference between an outcome and an event? Give an example to help clarify.

If you toss a coin, you cannot get both a head and a tail at the same time, so this has zero probability. An event that is certain to occur has a probability of 1. When you toss a coin, you will either get a Head or a Tail. Outcomes are the basic results of observations or experiments. Events are one or more outcomes that share a property of interest. With a two coin toss, there are 4 possible outcomes: HH, TT, HT, TH With a two coin toss, there are 3 events: HH TT and (HT & TH)

What is the probability of randomly meeting either a male or a sophomore in a small BYU-Idaho ward activity? Note, that at this ward activity, there are 10 sophomore females, 5 sophomore males, 28 freshmen females, and 18 freshmen males.

OR; overlapping 23/61 males + 15/61 sophomores - 5/61 (sophmores/males) = 33/61 or 54%

By definition, a millennial flood is a flood that is so severe that it only happens once every 1000 years on average. What is the probability that the area in which you live will have a millennial flood some time during the next decade? (Round to the nearest six decimals.)

P(1000 yr. flood) = 1/1000 P(no 1000 yr flood) = 999 At least once rule = 1 - (999/1000)^10 = .009955 1 - (probability of no millennial flood in a given year)10= 1 - (.999)10= 0.009955 or just a little less than 1%

So far this month your roommate has helped do the daily dishes 6 times. If it is the 12th day of the month, what is the chance (written as a percent) that your roommate will do the dishes today (assuming the previous pattern mentioned above)?

Since it's the 12th day of the month. I assume that the roommate did the dishes 6 times in the previous 11 days. 6/11 = .545454 or 54.4%

Find the reduced odds for and the reduced odds against the event of rolling a fair die and getting a 5.

odds for: 1 to 5 odds against 5 to 1 you have 1 out 5 to roll a 5 you have 5 out of 1 to not roll a 5

Make a table showing the probability distribution for the possible number of heads when tossing 5 coins.

we have p=q=1/2, and you want the case for n=5 which is given by: 1 = (p+q)5 = 1p5 + 5p4q + 10p3q2 + 10p2q3 + 5pq4 + 1q5 Because p=q=1/2, the pq parts of all of these terms are all the same: (1/2)5 = 1/32, so we have: 1 = (1 + 5 + 10 + 10 + 5 + 1) /32 The 6 different terms give the probabilites for 5 heads, 4 heads, ..., one head, zero heads. In a table: heads Probability 0 1/32 1 5/32 2 10/32 3 10/32 4 5/32 5 1/32