MATH 125 | Exam 3

3) A die is rolled. The set of equally likely outcomes is {1, 2, 3, 4, 5, 6}. Find the probability of getting a 10.

The probability of rolling a 10 on a standard die is 0/6, because none of the six possible outcomes (1–6) include 10.

A) 0

4) A card is drawn from a standard deck of 52 playing cards. Find the probability that the card is a picture card

The correct answer is 3/13, because there are 12 picture cards in a standard 52‑card deck (Jack, Queen, King in each of the four suits), and 12÷52 simplifies to 3/13.

A) 3/13

5) In the game of craps, two dice are tossed and the up faces are totaled.  Is the event getting a total of 9 and one of the dice showing a 6 mutually exclusive?  

A) No

6) A probability experiment is conducted in which the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. }. Let event A = {6, 7, 8, 9} and event B = {8, 9, 10, 11, 12}.  Assume that each outcome is equally likely. List the outcomes in A and B. Are A and B mutually exclusive?

A and B share the outcomes 8 and 9, so they are not mutually exclusive, which matches option A.

A) 8/9 No

7) A card is drawn from a standard deck of 52 playing cards. Find the probability that the card is a queen or a club. Express the probability as a simplified fraction.

The probability is 4/13 because there are 4 queens and 13 clubs, but one card is both, so after subtracting the overlap you get 16 favorable cards out of 52, which simplifies to 4/13.

A) 4/13

8) If one card is drawn from a standard 52 card playing deck, determine the probability of getting a ten, a king or a diamond. Round to the nearest hundredth.

The probability of drawing a ten, king, or diamond from a 52‑card deck is 0.37, because there are 4 tens, 4 kings, and 13 diamonds minus the 2 overlapping diamond cards, giving 19 favorable cards out of 52.

A)0.37

9) You are dealt one card from a 52-card deck. Find the probability that you are not dealt a 5. Express the probability as a simplified fraction.

The probability of not being dealt a 5 is 12/13, because there are 48 cards that are not fives out of 52 total cards, and the fraction 48/52 simplifies to 12/13.

A) 12/13

10) You are dealt one card from a 52 card deck.  Then the card is replaced in the deck, the deck is shuffled, and you draw again.  Find the probability of getting a picture card the first time and a club the second time. Express the probability as a simplified fraction.

The probability is 3/52, because there are 12 picture cards out of 52 for the first draw and 13 clubs out of 52 for the second draw, and multiplying these independent probabilities gives the simplified fraction 3/52.

A) 3/52

11) A human gene carries a certain disease from the mother to the child with a probability rate of 57%. That is, there is a 57% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has three children. Assume that the infections of the three children are independent of one another. Find the probability that all three of the children get the disease from their mother. Round to the nearest thousandth.

The probability is 0.185, because you multiply the mother’s infection rate for each child (0.57 × 0.57 × 0.57) and rounding that product to the nearest thousandth gives 0.185.

A) 0.815

12) Suppose a basketball player is an excellent free throw shooter and makes 94% of his free throws (i.e., he has a 94% chance of making a single free throw). Assume that free throw shots are independent of one another. Suppose this player gets to shoot three free throws. Find the probability that he misses all three consecutive free throws. Round to the nearest ten-thousandth.

The player misses a free throw 6% of the time, so the probability he misses all three is 0.06×0.06×0.06=0.000216, which rounds to 0.0002 to the nearest ten‑thousandth.

A) 0.0002

13) Computing the probability of the event ʺdrawing a second red ball from a bag of colored balls after having kept the red ball that was drawn first from the bagʺ is an example of

A) Conditional Probability

14) There are 35 chocolates in a box, all identically shaped. There 8 are filled with nuts, 12 with caramel, and 15 are solid chocolate. You randomly select one piece, eat it, and then select a second piece.  Find the probability of selecting a solid chocolate candy followed by a nut candy.

The correct answer is 12/119, because you multiply the chance of picking a solid chocolate first (15/35) by the chance of then picking a nut candy from the remaining chocolates (8/34), which simplifies to 12/119.

A)12/119

15) the number of bottles of juice sold in a cafeteria during lunch

The number of bottles of juice sold is discrete, because you can only count whole bottles and not fractions.

A) Discrete

16) the speed of a car on a New York tollway during rush hour traffic

B) Discrete

17) Given the table of probabilities for the random variable x, does this form a probability distribution?

x   5 10 15 20                     P(x) 0.10 -0.30 0.50 0.70 

Answer Yes or No.

one of the probabilities is negative, and probabilities must always be between 0 and 1.'

A) No

18) Consider the discrete probability distribution to the right when answering the following question. Find the probability that x equals 4.

x     3     4     6      8                           P(x)     0.04    ?     0.07    0.31

The probability that x=4 is 0.58 because all probabilities in a distribution must add up to 1, so you subtract the sum of the known probabilities (0.04+0.07+0.31) from 1 to get 0.58.

A) 0.58

19) Calculate the mean for the discrete probability distribution shown here.

x    2    5    9    10               P(x) 0.09   0.07    0.22    0.62

You multiply each value of x by its probability and then add all those products together.

A) 8.71

20) In a sandwich shop, the following probability distribution was obtained. The random variable x represents the number of condiments used for a hamburger. Find the mean and standard deviation for the random variable x.

x             P(x)

0             0.30

 1            0.40

2             0.20

3             0.06

4             0.04

The standard deviation is 1.04 because after finding the mean, you compute each squared deviation (x−μ)2, multiply each by its probability, add those results, and take the square root of that total.

A)Mean: 1.14; standard deviation: 1.04

21)  An inspector at a BMW plant is inspecting the number of BMW steering wheels that may have a defect. He inspects 10 steering sheels on a regular day. The probability that a steering wheel is defective Is determined to be 0.06. Find the probability that at least 4 have defects.

P(X≥4)≈0.002

21)  An inspector at a BMW plant is inspecting the number of BMW steering wheels that may have a defect. He inspects 10 steering sheels on a regular day. The probability that a steering wheel is defective

Is determined to be 0.06. Find the probability that at least 4 have defects.

22) Find the mean and standard deviation of the above scenario

• Mean: μ=0.6

• Standard deviation: σ≈0.75