Probability

True or false. Determine if the statements below are true or false, and explain your reasoning.

1. If a fair coin is tossed many times and the last eight tosses are all heads, then the chance that the next toss will be heads is somewhat less than 50%.

2. Drawing a face card (jack, queen, or king) and drawing a red card from a full deck of playing cards are mutually exclusive events.

3. Drawing a face…

Four games, one winner. Below are four versions of the same game. Your archnemesis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. Identify how many coin flips you should choose for each version of the game. It costs $1 to play each game. Explain your reasoning.

1. If the proportion of heads is larger than 0.60, you win $1.

2. If…

Coin flips. If you flip a fair coin 10 times, what is the probability of

1. getting all tails?

2. getting all heads?

3. getting at least one tails?

Disjoint vs. independent. In parts (a) and (b), identify whether the events are disjoint, independent, or neither (events cannot be both disjoint and independent).

1. You and a randomly selected student from your class both earn A’s in this course.

2. You and your class study partner both earn A’s in this course.

3. If two events can occur at the same time, must they be dependent?

Joint and conditional probabilities. P(A) = 0.3, P(B) = 0.7

1. Can you compute P(A and B) if you only know P(A) and P(B)?

2. Assuming that events A and B arise from independent random processes,

   1. what is P(A and B)?

   2. what is P(A or B)?

   3. what is P(A$|$B)?

3. If we are given that P(A and B) = 0.1, are the random variables giving rise to events A and B independent?

4. If we are given that P(A and B) =…

Global warming. A Pew Research poll asked 1,306 Americans “From what you’ve read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?”. The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies.

1. Are believing that the earth is warming and…

Marbles in an urn. Imagine you have an urn containing 5 red, 3 blue, and 2 orange marbles in it.

1. What is the probability that the first marble you draw is blue?

2. Suppose you drew a blue marble in the first draw. If drawing with replacement, what is the probability of drawing a blue marble in the second draw?

3. Suppose you instead drew an orange marble in the first draw. If drawing with replacem…

Chips in a bag. Imagine you have a bag containing 5 red, 3 blue, and 2 orange chips.

1. Suppose you draw a chip and it is blue. If drawing without replacement, what is the probability the next is also blue?

2. Suppose you draw a chip and it is orange, and then you draw a second chip without replacement. What is the probability this second chip is blue?

3. If drawing without replacement, what is the p…

College smokers. At a university, 13% of students smoke.

1. Calculate the expected number of smokers in a random sample of 100 students from this university.

2. The university gym opens at 9 am on Saturday mornings. One Saturday morning at 8:55 am there are 27 students outside the gym waiting for it to open. Should you use the same approach from part (a) to calculate the expected number of smokers a…

Hearts win. In a new card game, you start with a well-shuffled full deck and draw 3 cards without replacement. If you draw 3 hearts, you win $50. If you draw 3 black cards, you win $25. For any other draws, you win nothing.

1. Create a probability model for the amount you win at this game, and find the expected winnings. Also compute the standard deviation of this distribution.

2. If the game costs…

Portfolio return. A portfolio’s value increases by 18% during a financial boom and by 9% during normal times. It decreases by 12% during a recession. What is the expected return on this portfolio if each scenario is equally likely?

American roulette. The game of American roulette involves spinning a wheel with 38 slots: 18 red, 18 black, and 2 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money. S…

Grade distributions. Each row in the table below is a proposed grade distribution for a class. Identify each as a valid or invalid probability distribution, and explain your reasoning.

Cost of breakfast. Sally gets a cup of coffee and a muffin every day for breakfast from one of the many coffee shops in her neighborhood. She picks a coffee shop each morning at random and independently of previous days. The average price of a cup of coffee is $1.40 with a standard deviation of 30¢ ($0.30), the average price of a muffin is $2.50 with a standard deviation of 15¢, and the two prices…

Variance of a mean, Part I. Suppose we have independent observations $X_{1}$ and $X_{2}$ from a distribution with mean $\mu$ and standard deviation $\sigma$. What is the variance of the mean of the two values: $\frac{X_{1}+X_{2}}{2}$?

Variance of a mean, Part III. Suppose we have $n$ independent observations $X_{1}$, $X_{2}$, …, $X_{n}$ from a distribution with mean $\mu$ and standard deviation $\sigma$. What is the variance of the mean of these $n$ values: $\frac{X_{1}+X_{2}+\dots+X_{n}}{n}$?