## 25.1 Bernoulli trials

In a sequence of Bernoulli($$p$$) trials:

• There are only two possible outcomes, “success” (1) and “failure” (0), on each trial.
• The unconditional/marginal probability of success is the same on every trial, and equal to $$p$$.
• The trials are independent.

## 25.2 Binomial distributions

Example 25.1 In a random bit stream, the probability that any particular bit is 1 is 0.75, independently of all other bits. Let $$X$$ be the number of 1s in the next 10 bits.

1. Identify by name the distribution of $$X$$. Be sure to specify the values of relevant parameters.

2. Compute and interpret $$\text{P}(X=6)$$.

3. Compute and interpret $$\text{E}(X)$$.

• A discrete random variable $$X$$ has a Binomial distribution with parameters $$n$$, a nonnegative integer, and $$p\in[0, 1]$$ if its probability mass function is \begin{align*} p_{X}(x) & = \binom{n}{x} p^x (1-p)^{n-x}, & x=0, 1, 2, \ldots, n \end{align*} If $$X$$ has a Binomial($$n$$, $$p$$) distribution \begin{align*} \text{E}(X) & = np\\ \text{Var}(X) & = np(1-p) \end{align*}
• If $$X$$ counts the number of successes in a fixed number, $$n$$, of Bernoulli($$p$$) trials then $$X$$ has a Binomial($$n, p$$) distribution.

Example 25.2 In each of the following situations determine whether or not $$X$$ has a Binomial distribution. If so, specify $$n$$ and $$p$$. If not, explain why not.

1. Roll a die 20 times; $$X$$ is the number of times the die lands on an even number.

2. Roll a die 20 times; $$X$$ is the number of times the die lands on 6.

3. Roll a die until it lands on 6; $$X$$ is the total number of rolls.

4. Roll a die 20 times; $$X$$ is the sum of the numbers rolled.

5. Shuffle a standard deck of 52 cards (13 hearts, 39 other cards) and deal 5 without replacement; $$X$$ is the number of hearts dealt. (Hint: be careful about why.)

6. Roll a fair six-sided die 10 times and a fair four-sided die 10 times; $$X$$ is the number of 3s rolled (out of 20).

## 25.3 Geometric distributions

Example 25.3 Maya is a basketball player who makes 40% of her three point field goal attempts. Suppose that she attempts three pointers until she makes one and then stops. Let $$X$$ be the total number of shots she attempts. Assume shot attempts are independent.

1. Does $$X$$ have a Binomial distribution? Why or why not?

2. What are the possible values that $$X$$ can take? Is $$X$$ discrete or continuous?

3. Compute and interpret $$\text{P}(X=3)$$.

4. Find the probability mass function of $$X$$.

5. Construct a table, plot, and spinner representing the distribution of $$X$$.

6. Compute and interpret $$\text{P}(X>5)$$ without summing. (Hint: consider the first 5 trials.)

7. What seems like a reasonable general formula for $$\text{E}(X)$$? Make a guess, and then compute and interpret $$\text{E}(X)$$ for this example.

8. Would $$\text{Var}(X)$$ be bigger or smaller if $$p=0.9$$? If $$p=0.1$$?

9. Compute and interpret $$\text{P}(X > 15 | X > 10)$$. What do you notice?

• A discrete random variable $$X$$ has a Geometric distribution with parameter $$p\in[0, 1]$$ if its probability mass function is \begin{align*} p_{X}(x) & = p (1-p)^{x-1}, & x=1, 2, 3, \ldots \end{align*} If $$X$$ has a Geometric($$p$$) distribution \begin{align*} \text{E}(X) & = \frac{1}{p}\\ \text{Var}(X) & = \frac{1-p}{p^2} \end{align*}
• Suppose you perform Bernoulli($$p$$) trials until a single success occurs and then stop; let $$X$$ be the total number of trials performed, including the success. Then $$X$$ has a Geometric($$p$$) distribution. In this situation, exactly $$x$$ trials are performed if and only if
• the first $$x-1$$ trials are failures, and
• the $$x$$th (last) trial results in success.

## 25.4 Negative Binomial Distributions

Example 25.4 Maya is a basketball player who makes 86% of her free throw attempts. Suppose that she attempts free throws until she makes 5 and then stops. Let $$X$$ be the total number of free throws she attempts. Assume shot attempts are independent.

1. Does $$X$$ have a Binomial distribution? A Geometric distribution? Why or why not?

2. What are the possible values of $$X$$? Is $$X$$ discrete or continuous?

3. Compute and interpret $$\text{P}(X=5)$$

4. Compute and interpret $$\text{P}(X=6)$$

5. Compute and interpret $$\text{P}(X=7)$$

6. Compute and intepret $$\text{P}(Y=8)$$

7. Find the probability mass function of $$X$$.

8. What seems like a reasonable general formula for $$\text{E}(X)$$? Interpret $$\text{E}(X)$$ for this example.

9. Would the variance be larger or smaller if attempted free throws until she made 10 instead of 5?

• A discrete random variable $$X$$ has a Negative Binomial distribution with parameters $$r$$, a positive integer, and $$p\in[0, 1]$$ if its probability mass function is \begin{align*} p_{X}(x) & = \binom{x-1}{r-1}p^r(1-p)^{x-r}, & x=r, r+1, r+2, \ldots \end{align*} If $$X$$ has a NegativeBinomial($$r$$, $$p$$) distribution \begin{align*} \text{E}(X) & = \frac{r}{p}\\ \text{Var}(X) & = \frac{r(1-p)}{p^2} \end{align*}
• Suppose you perform a sequence of Bernoulli($$p$$) trials until $$r$$ successes occur and then stop. Let $$X$$ be the total number of trials performed, including the trials on which the successes occur. Then $$X$$ has a NegativeBinomial($$r$$,$$p$$) distribution. In this situation, exactly $$x$$ trials are performed if and only if
• there are exactly $$r-1$$ successes among the first $$x-1$$ trials, and
• the $$x$$th (last) trial results in success.
• There are $$\binom{x-1}{r-1}$$ possible sequences that satisfy the above, and each of these sequences — with $$r$$ successes and $$x-r$$ failures — has probability $$p^r(1-p)^{x-r}$$.

Example 25.5 What is another name for a NegativeBinomial(1,$$p$$) distribution?

Example 25.6 Suppose $$X_1, \ldots, X_r$$ are independent each with a Geometric($$p$$) distribution. What is the distribution of $$X_1+\cdots + X_r$$? Find the expected value and variance of this distribution.

Example 25.7 Each box of a certain type of cereal contains one of $$n$$ distinct prizes — labeled $$1, \ldots, n$$ — and you want to obtain a complete set. Suppose that each box of cereal is equally likely to contain any one of the $$n$$ prizes, independently from box to box. You purchase cereal boxes one box at a time until you have the complete set of $$n$$ prizes. Let $$X$$ be the number of boxes purchased to obtain a complete set of prizes. Compute and interpret $$\text{E}(X)$$ as a function of $$n$$.