Illinois State University Mathematics Department

 MAT 312: Probability and Statistics for Middle School Teachers Spring 1999 9:35 - 10:50 am TR STV 350A Dr. Roger Day (day@ilstu.edu)

 Possible Solutions to Quiz #4 look at grade sheet

 1. TRUE or FALSE: Theoretical probability considers all possible things that could happen in an experiment.
 2. TRUE or FALSE: An experiment consists of flipping three coins and recording the number of heads. The sample space for the experiment is the set {1,2,3}. The correct sample space is {0,1,2,3}.
 3. TRUE or FALSE: A fair die is rolled and the number on the face-up side is recorded, so the sample space is {1,2,3,4,5,6}. Each outcome in the sample space is equally likely.
 4. TRUE or FALSE: Two dice are rolled and the sum of the two face-up sides is recorded. Let event B represent that the sum is less than 5 and let event A represent that the sum is 7. Then P(B) = P(A). P(B) = P(2,3, or 4) = 6/36, and P(A) = P(7) = 6/36.
 5. TRUE or FALSE: Two dice are rolled and the sum of the two face-up sides is recorded. If event X represents that the sum is not 7 and event Y represents that the sum is greater than or less than 7, X and Y are complementary events. As described, X and Y are equivalent events, for they describe the same thing. The complement of X, symbolized ~X, is "the sum is 7."
 6. TRUE or FALSE: Two coins are flipped and the number of tails is recorded. If event A represents that no tails appear and event B represents that two tails appear, A and B are mutually exclusive events. Events A and B have nothing in common; they share no outcomes.
 7. SOMETIMES TRUE, ALWAYS TRUE, or NEVER TRUE: If two events are complementary, then they are mutually exclusive. What about the converse: If two events are mutually exclusive, then they are complementary?
 8. SOMETIMES TRUE, ALWAYS TRUE, or NEVER TRUE: An experiment consists of two tasks that involve a bag containing 3 green balls and 5 red balls. First, a ball is selected at random from the bag, the color of the ball is recorded, and the ball is placed in a waste can. Next, another ball is selected at random from the bag, the color of the ball is recorded, and this ball is placed in a waste can. If A represents the event "a red ball is selected" and B represents the event "a green ball is selected," then A and B are independent events. These events, for this situation, are never independent, for the probability of A or B depends on knowing when the draw is made and what the previous color was.
 9. SOMETIMES TRUE, ALWAYS TRUE, or NEVER TRUE: Two events with the property that P(X) + P(Y) = 1 are complementary events. We know the statement is at least sometimes true, for this is a property of complementary events. To show it is only sometimes true, we need an example for which the statement is false. Consider the experiment of rolling two fair dice and recording the sum on the two face-up sides. Let X be the event that the sum is no more than 9, and let Y be the event that the sum is 4 or less. Then P(X) = 30/36 and P(Y) = 6/36. We have P(X) + P(Y) = 1. The events are not complementary, however, for they share outcomes.
 10. SOMETIMES TRUE, ALWAYS TRUE, or NEVER TRUE: For an experiment with the sample space {0,1,2}, P(0) = P(1) = P(2). We show this is sometimes true by providing an example for which the statement is false and another example for which the statement is true. If the experiment is "flip two coins and record the number of heads" then the statement is false. If the experiment is "choose a number at random from the set {0,1,2}" then the statement is true.
 11. A single letter is drawn from the set {g,e,o,m,t,r,y}. What is the probability that the letter drawn is a member of the set {g,e,o} ? We assume each of the 7 letters in the first set are equally likely to be chosen. Then the probability of getting a letter from the fist set that is contained in the second set is 3/7.
 12. An experiment consists of rolling a single fair die. Determine the probability that the result is: a) greater than 3. P(4,5, or 6) = 3/6 = 1/2 c) a number divisible by 6. P(6) = 1/6 b) a multiple of 5. P(5) = 1/6 d) a number divisible by 3 and 5. P(div by 3 and by 5) = 0
 13. Two dice are rolled and the sum of the two face-up sides is recorded. What is the probability that one of the dice showed a "3" if we know that the sum was 7? This is a problem of conditional probability: P(one die shows 3 | sum is 7). We can solve this using the conditional probability formula or more informally by looking at the reduced sample space and counting. P(one die shows 3 | sum is 7) = P(one die shows 3 AND sum is 7)/P(sum is 7) = 2/36 / 6/36 = 2/6 = 1/3. If the sum is 7, there are 6 outcomes to look at: (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). Of these, two include a 3. Therefore, the desired probability is 2/6 = 1/3.
 14. Pat is about to take a 2-item multiple-choice quiz. Each item has six possible answers (a,b,c,d,e,f). Pat has no knowledge of the material and decides to roll a die to respond to each item. For multiple-choice question #1, if the die shows "1" Pat selects choice "a," if the die shows "2" Pat selects choice "b," and so on. Pat repeats this same die-rolling activity to choose a response for multiple-choice question #2. What is the probability that Pat gets both questions correct on this quiz? There are 36 different outcomes Pat could generate, from "a,a" through "f,f." Only one of these represents the pair of letters corresponding to the correct quiz responses. Therefore, the probability is 1/36.

15. An experiment consists of throwing two darts at the board shown to the right and recording the sum of the numbers showing within the regions hit by the darts.

For example, if one dart hits the upper-right sector of the circle and the other dart hits the lower-left sector, a sum of 8 is recorded. If both darts hit the lower-right sector, a sum of 16 is recorded.

Assume that each dart hits somewhere on the board and that each of the four sectors of the board represent one-fourth of a circle.

a) List the sample space for this experiment.

We first show all possible outcomes using a box array:

 + 2 4 6 8 2 4 6 8 10 4 6 8 10 12 6 8 10 12 14 8 10 12 14 16

The sample space is
{4,6,8,10,12,14,16}.

b) Determine P(12) for this experiment.

P(12) = 3/16.

c) Determine P(10 | one dart hits a 6).

P(10 | one dart hits a 6) = 2/7.

 Quiz Administered on Thursday 8 April 1999 .