Binomial tail probabilities for assessing claims and streaks

A sports columnist for the Dallas Morning News had a particularly bad week picking the winners of NFL football games-he got 1 right and 12 wrong, with 1 tie. Afterward, he wrote that, "Theoretically, a baboon at the Dallas Zoo can look at a schedule of 14 NFL games, point to 1 team for each game and come out with at least 7 winners." The next week, Kanda the Great, a gorilla at the Dallas Zoo, made his predictions by selecting pieces of paper from his trainer. Kanda got nine right and four wrong, better than all six Morning News sportswriters. Assuming no ties, what is the probability that a baboon picking the winners of 13 games will select at least 9?

Unanimous jury trials have historically been guaranteed in federal criminal cases, but not in state trials. In Ramos v. Louisiana (2019), the US Supreme Court considered a challenge to the constitutionality of a Louisiana law that allows a conviction if at least 10 of 12 jurors vote guilty. A Stanford law school professor argued that a unanimous verdict of 6 jurors is more trustworthy than a majority decision of 12 or even 20 jurors.

Suppose that each juror is randomly selected from a large pool, of which $90 \%$ of the potential jurors would vote guilty and $10 \%$ would vote not guilty. Compare the probability that 6 randomly selected jurors would all vote guilty with the probability that a majority of 20 jurors would vote guilty.

James Fishkin, director of the Center for Deliberative Democracy at Stanford, argued that, "There is so little time for deliberation that some people make leadership choices based on whether they like a candidate's hairstyle." Fishkin proposed that $200-300$ randomly selected citizens be asked to spend 1-2 days listening to experts debate a ballot initiative or candidate, and then vote. If $55 \%$ of all citizens would vote yes on a ballot initiative after listening to the experts, what is the exact probability that more than $50 \%$ of 250 randomly selected citizens would vote yes?

An investment manager wrote that, "If 1,000 people flip coins, some will get heads 10 times in a row, just by chance. You wouldn't anoint them superior coin flippers." If 1,000 people each flip ten coins,

A seemingly healthy woman has a physical checkup which involves a battery of tests of 20 risk factors (such as cholesterol) that might indicate a health problem. For each test, the result is flagged as abnormal if the reading is unusually high or low-specifically, if it is outside a range that encompasses $95 \%$ of the readings for healthy women. Thus, if a woman is healthy, there is only a $5 \%$ chance that her reading on a test will be outside the normal range. Assuming that the test results are independent, what is the probability that a healthy woman who takes 20 such tests will have two or more abnormal readings?

In 1980, the United States tried a daring rescue of American hostages that were being held by Iranians in the American Embassy in Teheran. The American plan was to bring in soldiers in eight helicopters traveling 800 miles across the desert at low altitudes during the night. The military commanders believed that at least six of the eight helicopters would have to reach Teheran for the mission to have a chance of succeeding. As it turned out, the rescue attempt was canceled when three of the helicopters became disabled. Assuming independence, calculate the probability that at least six of the eight helicopters would reach Teheran if each has a .75 probability of success. Why might independence not be a warranted assumption?

A basketball article in the Los Angeles Times was titled, "Wright Defies the Percentages, and USC Holds Off Oregon, 62-54." The article explained that "Oregon's strategy was obvious in the closing minutesfoul Gerry Wright, a $37.5 \%$ free throw shooter. But Wright, a reserve center, didn't fold under the pressure. He made 3 of his 6 foul shots in the final 2 minutes." If a player has a .375 probability of making a shot and each shot is independent, what is the probability of making at least three of six shots?

Each student admitted to ABC College has a. 6 probability of enrolling, and each student's decision is independent of the other students' decisions. Compare a college that admits 1,000 students with a college that admits 2,500 students. Which college has the higher probability that the percentage of students admitted who enroll will be

It was reported that a fair coin had been flipped in sixty ten-flip sets (a total of six hundred flips), and that, in two of the ten-flip sets, all ten flips had been heads and that, in another set, all ten flips had been tails. What is the probability that a fairly flipped coin would land either all heads or all tails in three or more of sixty ten-flip sets?

A marine biologist argued that when there is a long interval between calving, female whales are more likely to have sons than daughters. Data collected by the Center for Coastal Studies found that for whales with an interval of $1-2$ years between calving, there were 22 sons and 20 daughters. For mothers calving at intervals of 3,4, or 5 years, there were 16 sons and only 4 daughters. Assuming that these data are Bernoulli trials with a .5 probability of a son, what is the probability that of 20 calves, 16 or more will be male? (Do not use a normal approximation.)

In 2018, Eric Reid, a Carolina Panthers football player, said that the National Football League (NFL) was using its purportedly random drug-testing program to target him since he had been tested 7 times in 11 weeks. The NFL and NFL Players Association investigated and concluded that Reid's tests were random. His first test was a mandatory test after he signed with the Panthers. After that initial test, ten players on each NFL team are randomly selected each week. There are 72 players eligible for testing on the Carolina roster; assume that all 32 NFL teams have 72 players eligible for testing. If the testing is truly random, what is the probability that

A researcher tested for precognition by asking a volunteer to predict whether the next color shown on a computer screen will be red or green. The software used a random-event generator with a .5 probability of showing green and a. 5 probability of showing red. The volunteer got 60 out of 100 correct. Calculate the exact probability that a volunteer will get at least 60 right merely by guessing. If 25 guessing volunteers are tested, what is the probability that at least one will get 60 or more correct?

The Silvers are planning their family and want an equal number of boys and girls. Mrs. Silver says that their chances are best if they plan to have two children. Mr. Silver says that they have a better chance of having an equal number of boys and girls if they plan to have ten children. Assuming that boy and girl babies are equally likely and independent of previous births, which probability do you think is higher? Explain your reasoning.

A total of 98 college students were asked how many times they talked to their mothers and fathers during the past year: 36 talked to both parents equally; 9 talked to their fathers more; and 53 talked to their mothers more. If the 62 students who talked more to one parent are equally likely to say mother or father, what is the probability that fewer than 10 talk more to their father?

Mendel hypothesized that self-fertilization of hybrid yellow-seeded sweet peas would yield offspring with a .75 probability of being yellowseeded and a .25 probability of being green-seeded. In 1865, he reported that 8,023 such experiments yielded $6,021 / 8,023=.7505$ yellow-seeded plants and $2,002 / 8,023=.2495$ green-seeded plants. If these were honest independent trials, with each offspring having a .75 probability of being yellow-seeded, what is the exact probability that the number of yellow-seeded plants would be in the range of 6,013-6,021 (fewer than five plants from the expected value, $6,017.25$)?

A study of the anti-parasitic medication Ivermectin for COVID-19 was reported to have involved 600 patients, among whom 410 ages were even numbers and 190 ages were odd numbers. If odd and even ages are equally likely, what is the probability of such a large disparity?

A renowned stock picker claims that he has better than a $50 \%$ chance of picking stocks that will do better than the median stock over the next 12 months. To demonstrate this prowess, he picks ten stocks out of the Russell 3000 (which consists of the 3,000 largest US companies) and a year later it turns out that six have done better than the median return and four have done worse.

A statistics professors asked 14 students to imagine 10 coins flip and write down the sequence of heads and tails. Of the 14 students, 13 imagined sequences in which there were 4,5, or 6 heads. If each student had flipped a fair coin 10 times and written down the sequence that occurred, what is the probability that 13 or more students would have obtained sequences that had 4,5, or 6 heads?

Nate Silver's book The Signal and the Noise looks at five mortgages, each of which has a $5 \%$ chance of defaulting. Assuming independence, he calculates that probability that at least one will default as

What is the correct probability?

A radio station offered a $\$ 100$ prize to a listener who: (a) has a lucky one dollar bill with at least three 9s among the eight digits in the dollar's serial number (for example, 23944199 and 93944199 would be lucky dollars); and (b) is the correct caller, for example, the 47th person to call the radio station during the contest.

The random walk hypothesis says that the probability that stock prices will increase today is independent of past performance. If, on any given day, there is a .52 probability that the Dow Jones Industrial Average will go up and a .48 probability that it will go down, then (assuming independence) what is the probability that during the course of a year with 250 trading days, the Dow will have more up days than down days?

On average, $70 \%$ of the passengers on a flight from Los Angeles to Boston prefer chicken to fish. Assume that these preferences can be modeled as Bernoulli trials. If there are 200 passengers on a flight from LA to Boston and the airline carries 140 chicken dinners and 60 fish dinners, what is the probability that no one will be disappointed? That more than five passengers will be disappointed?

During a basketball game between Chicago and Detroit, the television commentator, Marv Albert, said that Michael Jordan was having trouble at the free-throw line, having made only 9 of 12 shots, while he had made $84 \%$ of his free throws going into the game. Assuming the Bernoulli trial model applies, what is the exact probability of 9 or fewer successes in 12 trials, each with a .84 success probability?

A high school basketball coach who was named "coach of the decade" by the Los Angeles Times told his players that missed shots bounce long (away from the shooter) $60 \%$ of the time. A study of the video tapes of three men's Division III basketball games found that 84 of 169 missed shots bounced short and 85 bounced long. Assuming that the Bernoulli trial model applies with a probability p that a missed shot will bounce long, determine whether \mathbb {P}\left(\mathrm{X}=85 if \mathrm{p}=.6\right) or \mathbb {P}\left(\mathrm{X}=85 if \mathrm{p}=.5\right) is larger.

In the board game, Settlers of Catan, players take turns rolling a pair of normal six-sided dice, with every player eligible to receive "resources" based on the number rolled. In one recent game, three of the first six rolls were 3 (a 1 on one die and a 2 on the other die). What are the chances of three or more 3s in the next six rolls? If you play Settlers 500 times, what is the probability that at least one game will begin with three or more 3s on the first six rolls?