4.7: Additional Exercises

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Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
Rio Hondo College

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Exploration

Some of these questions were adapted from puzzles at mindyourdecisions.com.

A small college has been accused of gender bias in its admissions to graduate programs.
1. Out of 500 men who applied, 255 were accepted. Out of 700 women who applied, 240 were accepted. Find the acceptance rate for each gender. Does this suggest bias?
2. The college then looked at each of the two departments with graduate programs, and found the data below. Compute the acceptance rate within each department by gender. Does this suggest bias?
3. Looking at our results from Parts a and b, what can you conclude? Is there gender bias in this college’s admissions? If so, in which direction?
A bet on “black” in Roulette has a probability of 18/38 of winning. If you win, you double your money. You can bet anywhere from $1 to $100 on each spin.
1. Suppose you have $10, and are going to play until you go broke or have $20. What is your best strategy for playing?
2. Suppose you have $10, and are going to play until you go broke or have $30. What is your best strategy for playing?
Your friend proposes a game: You flip a coin. If it’s heads, you win $1. If it’s tails, you lose $1. However, you are worried the coin might not be fair coin. How could you change the game to make the game fair, without replacing the coin?
Fifty people are in a line. The first person in the line to have a birthday matching someone in front of them will win a prize. Of course, this means the first person in the line has no chance of winning. Which person has the highest likelihood of winning?
Three people put their names in a hat, then each draws a name, as part of a randomized gift exchange. What is the probability that no one draws their own name? What about with four people?
How many different “words” can be formed by using all the letters of each of the following words exactly once?
1. “ALICE”
2. “APPLE”
How many different “words” can be formed by using all the letters of each of the following words exactly once?
1. “GRUMPS”
2. “TEETER”
The Monty Hall problem is named for the host of the game show Let’s make a Deal. In this game, there would be three doors, behind one of which there was a prize. The contestant was asked to choose one of the doors. Monty Hall would then open one of the other doors to show there was no prize there. The contestant was then asked if they wanted to stay with their original door, or switch to the other unopened door. Is it better to stay or switch, or does it matter?
Suppose you have two coins, where one is a fair coin, and the other coin comes up heads 70% of the time. What is the probability you have the fair coin given each of the following outcomes from a series of flips?
1. 5 Heads and 0 Tails
2. 8 Heads and 3 Tails
3. 10 Heads and 10 Tails
4. 3 Heads and 8 Tails
Suppose you have six coins, where five are fair coins, and one coin comes up heads 80% of the time. What is the probability you have a fair coin given each of the following outcomes from a series of flips?
1. 5 Heads and 0 Tails
2. 8 Heads and 3 Tails
3. 10 Heads and 10 Tails
4. 3 Heads and 8 Tails
We’ll now consider a simplification of a case from the 1960s. In the area, about 26% of the jury eligible population was black. In the court case, there were 100 men on the juror panel, of which 8 were black. Does this provide evidence of racial bias in jury selection?
In this problem, we will explore probabilities from a series of events.
1. If you flip 20 coins, how many would you expect to come up “heads”, on average? Would you expect every flip of 20 coins to come up with exactly that many heads?
2. If you were to flip 20 coins, what would you consider a “usual” result? An “unusual” result?
3. Flip 20 coins (or one coin 20 times) and record how many come up “heads”. Repeat this experiment 9 more times. Collect the data from the entire class.
4. When flipping 20 coins, what is the theoretic probability of flipping 20 heads?
5. Based on the class’s experimental data, what appears to be the probability of flipping 10 heads out of 20 coins?
6. The formula $_{n} C_{x} p^{x}(1-p)^{n-x}$ will compute the probability of an event with probability $p$ occurring $x$ times out of $n$, such as flipping $x$ heads out of $n$ coins where the probability of heads is $p = \frac{1}{2}$. Use this to compute the theoretic probability of flipping 10 heads out of 20 coins.
7. If you were to flip 20 coins, based on the class’s experimental data, what range of values would you consider a “usual” result? What is the combined probability of these results? What would you consider an “unusual” result? What is the combined probability of these results?