Probabilities

Flip a coin 200 times (use a real coin, not a computer simulation). Divide your results into 20 groups of 10 flips, and count the number of heads in each of these 20 groups. How often did you get ten heads in ten flips? Nine heads? And so on. Use a histogram to summarize your results. Now identify the longest streak of consecutive heads or consecutive tails in each of these 20 groups. How often did you get a streak of ten? A streak of nine? And so on. Use a histogram to summarize your results.

Baseball-reference.com is one source of detailed historical statistics for Major League Baseball (MLB) players. Select a completed season and identify the players who had 502 times at bat that season (the minimum required to qualify for season batting awards). Find the batting averages and wins above replacement (WAR) for each of these players. Display your data in histograms and calculate the means and standard deviations. What fractions of the batting averages and WARs are within one, two, and three standard deviations of the mean?

Baseball-reference.com is one source of detailed historical statistics for MLB players. Select a completed season and identify the players who pitched at least 162 innings that season (the minimum required to qualify for season pitching awards). Find the earned run averages (ERA) and WAR for each of these players. Display your data in histograms and calculate the means and standard deviations. What fractions of the ERAs and WARs are within one, two, and three standard deviations of the mean?

Baseball-reference.com is one source of detailed historical statistics for MLB players. One way to compare outstanding players from different historical periods is to see how many standard deviations each player's performance was from the average performance that year. Considering only the batting averages and WAR of players who had at least 502 times at bat that season (the minimum number required to qualify for season batting awards), compare Miguel Cabrera's performance in 2015, Tony Gwynn's performance in 1996, and Ted Williams' performance in 1941.

Baseball-reference.com is one source of detailed historical statistics for MLB players. One way to compare outstanding players from different historical periods is to calculate how many standard deviations each player's performance was from the average performance that year. Considering only the ERA and WAR of players who pitched at least 162 innings that season (the minimum required to qualify for season pitching awards), compare Zack Greinke's performance in 2015, Greg Maddux's performance in 1995, and Sandy Koufax's performance in 1966.

Identify the two players in the American League and the two players in the National League who hit the most home runs in the most recently completed MLB season. For each of these players, use the data for the three most recent seasons to estimate the probability p that this player will hit a home run during a time at bat. Also use your data for each of these four players over all three seasons to estimate the average number of times at bat n during a season. Now, assuming the binomial model is appropriate, use your estimates of p and n to estimate each player's chances of hitting more than Aaron Judge's (non-steroid-assisted) 2022 record of 62 home runs in a season.

Stock returns are the product of returns over shorter horizons. The logarithm of returns is the sum of the logarithm of returns over shorter horizons and may be governed by the normal distribution. Roll two standard six-sided dice 200 times and record the product of the numbers on the two dice. Now calculate the natural logarithm of each of these 200 products and display your results in a histogram.

You offer a friend a choice of three bags-one containing a candy bar or some other desirable prize and the other two bags containing already-been-chewed (ABC) gum. You know which bag contains the good prize, but your friend does not. After your friend picks a bag (and before looking inside), show your friend a bag that was not picked and contains ABC gum and ask your friend if they want to change their choice. Play this game with a large number of friends and record: (a) how often people switch; and (b) how often they would have won if they had switched every time.

Mr. Smith is walking with his daughter and has a second child at home. It has been argued that the probability that the child at home is a girl is either one-third or one-half. What do you think? Gather your own data by asking a large number of people how many siblings they have. If they have one sibling, record the sex of the person and the sibling. Is the frequency with which the sexes are the same closer to one-third or one-half?

You have three cards: one black on both sides, one white on both sides, and one white on one side and black on the other side. After the three cards are shuffled and dropped into an empty bag, you pull one card out and look at one side of the card. What is the probability that the back of the card is the same color as the front of the card? Play this game a large number of times and see if your results confirm or contradict your expectations.