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 Sample Quiz #4

1. TRUE or FALSE: Experimental probability considers all possible things that could happen in an experiment.

 a) TRUE b) FALSE The description above is of theoretical probability, for it considers what could happen rather than what we know has happened.

2. TRUE or FALSE: An experiment consists of flipping a coin and recording the result. The sample space for the experiment is the set {heads, tails}.

 a) TRUE b) FALSE

3. TRUE or FALSE: Two coins are flipped and the number of heads showing is recorded, so the sample space is {0,1,2}. Each outcome in the sample space is equally likely.

 a) TRUE b) FALSE Using a table, tree diagram, or other representation, we can show that there are four outcomes to this experiment, where H represents the result head and T represents the result tail: HH, HT, TH, TT. Each of these four outcomes is equally likely. For the sample space {0,1,2}, however, as described in the question, the outcomes are not equally likely because there are two ways to get 1 head and just one way to get either 2 heads or 2 tails.

4. TRUE or FALSE: Two dice are rolled and the sum of the two face-up sides is recorded. Let event A represent that the sum is greater than 8. Then P(A) = 4/11.

 a) TRUE b) FALSE Return to the notes from Tuesday, 30 March, to look at the sample space for this experiment. We see that of the 36 possible outcomes to this experiment, 10 of them meet the condition stated: the sum is greater than 8. Therefore, P(A) = 10/36 = 5/18. The incorrect response of 4/11 was likely determined by assuming the 11 outcomes are equally likely, and that four of the 11 outcomes satisfy the required condition.

5. TRUE or FALSE: Two dice are rolled and the sum of the two face-up sides is recorded. If event A represents that the sum is an even number and event B represents that the sum is an odd number, A and B are complementary events.

 a) TRUE b) FALSE Two conditionas are met here, assuring us that the two events are complementary. The two events cover all the outcomes in the sample space, for a sum can only be odd or even. Second, the two events have nothing in common, for a particular sum cannot be both odd and even.

6. TRUE or FALSE: Two dice are rolled and the sum of the two face-up sides is recorded. If event A represents that the sum is an even number and event B represents that the sum is an odd number, A and B are mutually exclusive events.

 a) TRUE b) FALSE This is true for the second reason stated for question (5) above: The two events have nothing in common, for a particular sum cannot be both odd and even.

7. If two events are mutually exclusive, then they are complementary.

 a) SOMETIMES TRUE b) ALWAYS TRUE c) NEVER TRUE From questions (5) and (6) above, we know that this is at least sometimes true, for we have one example. To show it is not always true, we need an example where it is not true. Try this for the same experiment as used in questions (5) and (6): Let M be the event that a 3 is rolled and let N be the event that a 4 or 6 is rolled. Then M and N are mutually exclusive, for the events have nothing in common. The two events are not complementary, however, for the two events do not include all possible outcomes of the experiment. Now convince yourself that the converse of the above statement, If two events are complementary, then they are mutually exclusive, is ALWAYS TRUE.

8. An experiment consists of two tasks. First, a fair coin is tossed and the result is recorded. Next, a die is rolled and the value on the face-up side is recorded. If A represents the event "a head results" for the first task and B represents the event "an even number is recorded" for the second task, then A and B are independent events.

 a) SOMETIMES TRUE b) ALWAYS TRUE c) NEVER TRUE These are independent events and always will be because the result of the coin toss has no influence on the result of the die roll.

9. For events X and Y, P(X or Y) is greater than P(X and Y). Assume that X and Y do not represent the same event and that events X and Y are not impossible events.

 a) SOMETIMES TRUE b) ALWAYS TRUE c) NEVER TRUE This is always true because of the inclusive nature of the OR conjunction and the exclusive nature of the AND conjunction. For the event X OR Y, we must have an outcome in X or in Y, but not necessarily in both. For the event X AND Y, an outcome must be in both X and in Y. As long as X and Y represent different events, it always will be the case that the OR conjunction includes more outcomes than the AND conjunction. Here's an example for you to work thruogh to help illustrate this. Suppose a fair die is rolled and the face-up result is recorded. The sample space is {1,2,3,4,5,6}. Let event X represent that the outcome is an odd number and let event Y represent that the outcome is a prime number. You take it from here!

10. For events R and T, to calculate P(R|T) we must know P(T).

 a) SOMETIMES TRUE b) ALWAYS TRUE c) NEVER TRUE Some will argue that this is always true because P(R|T) is the conditional probability of event R given that event T has occurred. The formula for P(R|T) is P(R|T) = P(R and T)/P(T). Some may argue that this is never true. Consider the experiment of rolling a fair die. Suppose we want to determine the conditional probability that a 3 results, given that the result is an odd number. We can reason that the probability is 1/3. By knowing that an odd number resulted, we can narrow our sample space to {1,3,5}, each equally likely to occur. Now we ask the probability of getting a 3 for this new (restricted) sample space. There is one way out of the three to get a 3, so the probability is 1/3. I did not have to know the probability of getting an odd number to calculate this probability. While the second argument may be compelling, how would you answer the following question if I hadn't provided P(T)? If P(R and T) = 0.25 and P(T) = 0.5, determine P(R|T).

 11. A single letter is drawn from the set {g,e,o}. What is the probability that the letter drawn is a member of the set {g,e,o,m,t,r,y}? The desired probability is 1, because the set {g,e,o} is entirely contained in the set {g,e,o,m,t,r,y}. No matter what letter is drawn from the first set, it it a sure thing that the letter will be in the second set.
 12. An experiment consists of rolling a single fair die. Determine the probability that the result is: a) greater than or equal to 5: 2/6 = 1/3 b) an odd number: 3/6 = 1/2 c) a number divisible by 2 or 3: 4/6 = 2/3 d) a number divisible by 2 and 3: 1/6
 13. A coin is flipped three times and the number of heads is recorded. a) List the sample space for this experiment: {0,1,2,3} b) Determine the probability of each unique outcome in the sample space. P(0) = 1/8, P(1) = 3/8, P(2) = 3/8, P(3) = 1/8
 14. An experiment has six mutually exclusive outcomes, call them A, B, C, D, E, and F. If the sample space for the experiment is a uniform sample space, what is P(A or D)? Beacuse the sample space is uniform and the events are mutually exclusive, we know all outcomes are equally likely and each is equal to 1/6. This means we add the probabilities of the individual outcomes to determine P(A or D), resulting in a probability of 1/3.

15. Pat is about to take a 4-intem TRUE/FALSE quiz. Pat has no knowledge of the material and decides to flip a coin to answer each item. What is the probability that Pat earns a perfect score on the quiz?

There are 16 outcomes possible when a coin is flipped four times in succession and the results recorded:

 HHHH HTHH THHH TTHH HHHT HTHT THHT TTHT HHTH HTTH THTH TTTH HHTT HTTT THTT TTTT

If we think of H as TRUE and T as FALSE, these represent thd 16 ways Pat could have randomly chosen responses on the quiz. Of course, only one of these arrangements represents a perfect response from Pat. The desired probability is therefore 1/16.

 Actual Quiz to be Administered Thursday 8 April 1999 .