Practice problems‐probability and probability distributions. In problems 1‐3, explicitly write out the sample space, and calculate the probability of the indicated events. 1. You toss a fair coin 4 times. a. What is the probability the coin lands heads all 4 times? b. What is the probability the coin lands heads exactly twice? c. What is the probability there is either a pair of consecutive heads or a pair of consecutive tails? 2. A professor teaches a very small class with only 4 students. One day when returning a test, he simply shuffles the tests and gives each student one of the 4 tests at random. a. What is the probability all 4 students receive the correct test? b. What is the probability that no one receives the correct test? c. What is the probability that one or more of the students receives the correct test? 3. You have two six sided dice. However, each die has been modified so that instead of 1,2,3,4,5,6, the sides are numbered 1,2,2,2,3,4. When you throw the two dice… a. What is the probability the total of the two dice is 4? b. What is the probability the total of the two dice is an even number? 4. Explain in your own words the difference between objective and subjective probabilities, giving examples of each. In problems 5‐7, determine if the events in question are independent or not. 5. You throw two fair coins. a. A = “coin 1 lands heads”, B = “coin 2 lands tails” b. A = “coin 1 lands heads”, B = “both coins land heads” 6. See problem 3. A = “die #1 shows 2”, B = “the total of the two dice is 5”. 7. (Based on a true story) A professor observes two students having a conversation. Student #1: How’d you do on the first exam? Student #2: I got a 25, but I figure as long as I get 100 on the next two exams and the final, I’ll be fine. Professor overhearing the conversation: a. If A = “student 2 got 25 on the first exam” and B = “student 2 will get 100 on everything else in the course”, are A and B independent? b. Does student 2 believe that A and B are independent? 8. A couple intends to have two children. It’s estimated that approximately 52% of births are male, and 48% are female. Find the probabilities that both children are male, both children are female, and the probability they have one of each. 9. You have a shuffled 52 card deck, and you deal yourself two cards. a. What is the probability you deal yourself two aces? b. What is the probability you deal yourself a pair (two cards of the same rank)? c. What is the probability you deal yourself two cards of the same suit? 10. Baseball’s Division Series is a best of five game series. That is, the first team to win 3 games is the winner. Two teams (A and B) are playing in the series, and A is the clear favorite. In fact, you believe A will defeat B in any given game with probability .65, and each game is independent. a. What is the probability A wins the series? Hint: start by writing out all the different ways A could win. For instance AAA (A wins in three) is one possibility, ABAA (A wins in four, dropping game 2) is another, BAAA is another, etc. b. What if the two teams played a best‐of‐three series? 11. If the New England Patriots get home‐field advantage, you believe there is a 60% probability they will go to the Super Bowl. If not, this probability is only 30%. Assuming a 70% probability that the Pats get home‐field advantage, what is the probability they will go to the Super Bowl? 12. See problem 3. What is the probability both dice show the same number when thrown? Do the problem twice‐once using the sample space and once using a tree diagram. 13. A certain genetic disease is caused by the presence of two recessive alleles. That is, AA’s do not have the disease, Aa’s show no symptoms but are carriers, and aa’s are diseased. You believe that 81% of the population is AA, 18% is Aa, and 1% is aa. Then, imagine pairing off members of the population at random, and each pairing produces exactly one offspring. What percentage of the next generation will be AA, Aa, and aa respectively? Part of the tree diagram is given below. Male Female
Child
AA AA AA Aa Aa Plus many more rows… AA
Aa
AA
AA
Aa
Aa
AA
Aa
AA
Aa
Aa
aa
AA
Aa
Probability of this branch 14. Students wishing to be Mathematics majors are generally placed into either Precalculus, Calculus I or Calculus II as their first college Math course. Among students placing into Precalculus, 20% eventually complete the math major. For students placing into Calculus I and Calculus II, this number is 50% and 60% respectively. If the percentage of prospective Math majors placing into Precalculus, Calculus I and Calculus II is 40%, 50%, and 10% respectively, what percentage of them complete the Math major? 15. On the TV show Hell’s Kitchen, a group of chefs attempt to complete a dinner service under the exacting supervision of chef Gordon Ramsay. One evening, Ramsay hands out comment cards to the diners and asks their opinion of the food and whether or not they would be willing to pay $75 for the meal they were served. The data is below. Rating of food Excellent (E) Good (G) Dog’s Dinner (D)
Would pay?
Yes (Y)
40
25
1
No (N) 5
15
14
If you pick a diner at random from this population, compute… a. P(E), the probability the diner found the food excellent. b. P(E or Y), the probability the diner found the food excellent, or they would pay $75 for the meal c. P(D|N) d. P(N|D) e. Compute P(E and Y) and |
16. A university has three colleges, and they keep track of how many of the male/female applicants to each college are admitted. There was once a famous court case where female students claimed that they were being discriminated against, since a lower percentage of female students were admitted than male students. However, the university claimed that within each of its three colleges, the percentage of female students admitted was higher than the percentage of males admitted. Further, both of these claims were found to be true. Give an example to show how this is possible. 17. See problem #3. When you throw a pair of these modified dice, let X = the total appearing on the two dice. Fill in the chart below, and then find the mean of the random variable X. X
2
3
4
5
6
7
8
P(X)
18. A few years ago, Bill Belichick of the New England Patriots made a controversial call in a game against Peyton Manning’s Colts. The Patriots were facing 4th down and 1 yard to go at their own 28 yard line leading 34‐28 with 2:08 left in the fourth quarter. The Patriots failed to convert, and Peyton Manning promptly threw a TD pass to give the Colts a 35‐34 win. The question is whether Belichick’s decision was correct. As any poker player (or anyone who understands random phenomena) will tell you, it is possible to make all the correct decisions and still lose. His two options on 4th down were to (1) Go for the 1st down. At the least, if the Patriots made the first down, they have 1st and 10 at their own 29 with about 2:00 left. In this event it is estimated (through NFL game data and simulation‐see http://wp.advancednflstats.com/winprobcalc1.php if you’re interested) that the Patriots’ probability of winning the game is approximately 93%. If they fail to convert, the Colts have the ball at the Patriots’ 28 with 2:00 left down by 6 points. In this case, teams in this situation win about 32% of the time (meaning there would still be a 68% probability of a Patriots win even if they fail to convert the 4th down). (2) Punt. Assuming a net 40 yard punt, this gives the Colts the ball 1st and 10 at about their own 30 with 2:00 left. Teams in this situation win about 26% of the time (meaning there would be a 74% probability of a Patriots win in this case). a. If Belichick thought there was a 70% chance that his 4th down conversion attempt would succeed, was going for it the right play? That is, did going for it on 4th down give the Patriots a better probability of winning than punting? b. If p = the probability of converting the 4th down attempt, what is the minimum p that makes going for it on 4th down a better play than punting? 19. You sit down to take the final exam in one of your courses. Based on how you’ve prepared for the exam, you believe that your final grade X in the course follows the shown probability distribution. X
4 (A)
3 (B)
2 (C)
1 (D)
0 (F)
P(X)
.70
.20
.08
.01
.01
Find the mean of X. 20. See problem #19. You are taking two classes (perhaps it’s a summer term), and you believe your grades in each of the two classes are independent and have the probability distribution given in problem 19. Then, write down the probability distribution for X = your GPA in the summer term. Here are the first couple entries. To get a 4.0, both classes must be A’s. This would happen with probability . 7
.7
.49. You could also get a 3.5. This could happen either by getting an A in class #1 and a B in class #2, or vice versa. So, this would happen with probability . 7 .2 .2 .7 .28. Then find the mean of X.