*68*

TheÂ **binomial distribution**Â is one of the most commonly used distributions in statistics. This tutorial explains how to use the following functions in Excel to solve questions about binomial probabilities:

**BINOM.DIST****BINOM.DIST.RANGE****BINOM.INV**

**BINOM.DIST**

The functionÂ **BINOM.DISTÂ **finds the probability of getting a certain number of*Â *successesÂ in a certain number of trials where the probability of success on each trial is fixed.

The syntax forÂ **BINOM.DISTÂ **is as follows:

**BINOM.DIST**(number_s, trials, probability_s_cumulative)

**number_s:**Â number of successes**trials:Â**total number of trials**probability_s:Â**probability of success on each trial**probability_s_cumulative:**TRUE returns the cumulative probability; FALSE returns the exact probability

The following examples illustrate how to solve binomial probability questions using **BINOM.DIST**:

**Example 1**

*Nathan makes 60% of his free-throw attempts. If he shoots 12 free throws, what is the probability that he makes exactly 10?*

To answer this question, we can use the following formula in Excel: **BINOM.DIST(10, 12, 0.6, FALSE)**

The probability that Nathan makes exactly 10 free throw attempts out of 12 isÂ **0.063852**.

**Example 2**

*Marty flips a fair coin 5 times. What is the probability that the coin lands on heads 2 times or fewer?*

To answer this question, we can use the following formula in Excel: **BINOM.DIST(2, 5, 0.5, TRUE)**

The probability that the coin lands on heads 2 times or fewer isÂ **0.5**.

**Example 3**

*Mike flips a fair coin 5 times. What is the probability that the coin lands on heads more than 3 times?*

To answer this question, we can use the following formula in Excel:Â **1 â€“**Â **BINOM.DIST(3, 5, 0.5, TRUE)**

The probability that the coin lands on heads more than 3 times isÂ **0.1875**.

**Note:**Â In this example, BINOM.DIST(3, 5, 0.5, TRUE) returns the probability that the coin lands on heads 3 times or fewer. So, to find the probability that the coin lands on heads more than 3 times, we simply use 1 â€“ BINOM.DIST(3, 5, 0.5, TRUE).

**BINOM.DIST.RANGE**

The functionÂ **BINOM.DIST.RANGEÂ **finds the probability of getting a certain number of*Â *successesÂ in a certain range, based on a certain number of trials where the probability of success on each trial is fixed.

The syntax forÂ **BINOM.DIST.RANGEÂ **is as follows:

**BINOM.DIST.RANGE**(trials, probability_s, number_s, number_s2)

**trials:Â**total number of trials**probability_s:Â**probability of success on each trial**number_s:**Â minimum number of successes**number_s2:**Â maximum number of successes

The following examples illustrate how to solve binomial probability questions using **BINOM.DIST.RANGE**:

**EXAMPLE 1**

*Debra flips a fair coin 5 times. What is the probability that the coin lands on heads between 2 and 4 times?*

To answer this question, we can use the following formula in Excel:Â **BINOM.DIST.RANGE(5, 0.5, 2, 4)**

The probability that the coin lands on heads between 2 and 4 times isÂ **0.78125**.

**EXAMPLE 2**

*It is known that 70% of men support a certain law. If 10 men are randomly selected, what is the probability that between 4 and 6 of them support the law?*

To answer this question, we can use the following formula in Excel:Â **BINOM.DIST.RANGE(10, 0.7, 4, 6)**

The probability that between 4 and 6 of the randomly selected men support the law is **0.339797**.

**EXAMPLE 3**

*Teri makes 90% of her free-throw attempts. If she shoots 30 free throws, what is the probability that she makes between 15 and 25?*

To answer this question, we can use the following formula in Excel:Â **BINOM.DIST.RANGE(30, .9, 15, 25)**

The probability that she makes between 15 and 25 free throws is **0.175495**.

**BINOM.INV**

The functionÂ **BINOM.INVÂ **finds the smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value.

The syntax forÂ **BINOM.INVÂ **is as follows:

**BINOM.INV**(trials, probability_s, alpha)

**trials:Â**total number of trials**probability_s:Â**probability of success on each trial**alpha:**Â criterion value between 0 and 1

The following examples illustrate how to solve binomial probability questions using **BINOM.INV**:

**EXAMPLE 1**

*Duane flips a fair coin 10 times. What is the smallest number of times the coin could land on heads so that the cumulative binomial distribution is greater than or equal to 0.4?*

To answer this question, we can use the following formula in Excel:Â **BINOM.INV(10, 0.5, 0.4)**

The smallest number of times the coin could land on heads so that the cumulative binomial distribution is greater than or equal to 0.4 isÂ **5**.

**EXAMPLE 2**

*Duane flips a fair coin 20 times. What is the smallest number of times the coin could land on heads so that the cumulative binomial distribution is greater than or equal to 0.4?*

To answer this question, we can use the following formula in Excel:Â **BINOM.INV(20, 0.5, 0.4)**

The smallest number of times the coin could land on heads so that the cumulative binomial distribution is greater than or equal to 0.4 is **9**.

**EXAMPLE 3**

*Duane flips a fair coin 30 times. What is the smallest number of times the coin could land on tails so that the cumulative binomial distribution is greater than or equal to 0.7?*

To answer this question, we can use the following formula in Excel:Â **BINOM.INV(20, 0.5, 0.4)**

The smallest number of times the coin could land on tails so that the cumulative binomial distribution is greater than or equal to 0.7 is **16**.