Alta - Discrete Random Variables Chapter 4 Exam Questions with Verified Solutions (Rated 100%)
Burt, a football quarterback, has a pass completion percentage of 55.2%. If the probability that Burt will
need 8 or more pass attempts to complete his first pass in a game is at least 0.01, then Burt will be
benched and replaced by a backup. Will Burt be benched? Use Excel to find the probability, rounding to
three decimal places. - Answers Burt will not be benched since the probability is less than 0.01.
The probability that Burt will need eight or more attempts to complete his first pass is the complement
of the probability that he will need at most seven attempts. Use Excel to find the probability. Let a
success be Burt completing a pass.
1. First press FORMULAS and then INSERT FUNCTION.
2. Then select the NEGBINOM.DIST function.
3. Next enter the values for the number of failures before the first success, the number of successes,
and the probability of a success. In this case, since we want to determine the probability of at most
seven trials, there are at most six failures before the first success. Thus, enter 6, 1, and 0.552, in that
order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability.
4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.996378, which
is 0.996 rounded to three decimal places.
The complement of this probability is 1−0.996=0.004, which is less than 0.01. So Burt will not be
benched.
In a recent basketball season, Jenny sunk a three-point shot once in every 3.5 attempts, on average.
Assume that this probability did not change going into the next season. What is the probability that
Jenny sinks her first three-point shot on her third attempt of the season? Use Excel to find the
probability.
Round your answer to three decimal places. - Answers $0.146$0.146
Use Excel to find the probability. Note that this is not a cumulative probability. Let a success be Jenny
sinking a three-point shot.
1. First press FORMULAS and then INSERT FUNCTION.
2. Then select the NEGBINOM.DIST function.
3. Next enter the values for the number of failures before the first success, the number of successes,
and the probability of a success. In this case, since we want to determine the probability of three trials,
there are 2 failures before the first success. Thus, enter 2, 1, and 1/3.5, in that order. In the entry for a
cumulative probability, enter 0 since this is not a cumulative probability.
4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.145773, which
is 0.146 rounded to three decimal places.
,In a recent baseball season, Bob hit a home run approximately once every 18.38 plate appearances.
Assume that this probability did not change going into the next season. What is the probability that Bob
hits his first home run before his 25th plate appearance of the season? Use Excel to find the probability.
Round your answer to three decimal places. - Answers $0.739$0.739
Use Excel to find the probability. Note that this is a cumulative probability. Let a success be hitting a
home run.
1. First press FORMULAS and then INSERT FUNCTION.
2. Then select the NEGBINOM.DIST function.
3. Next enter the values for the number of failures before the first success, the number of successes,
and the probability of a success. In this case, since we want to determine the probability of at most 24
trials, there are at most 23 failures before the first success. Thus, enter 23, 1, and 1/18.38, in that order.
In the entry for a cumulative probability, enter 1 since this is a cumulative probability.
4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.738843, which
is 0.739 rounded to three decimal places.
An urn contains 4 total balls, which comprise 3 white balls and one green ball. Dwight is running an
experiment where he draws a ball from the urn, records the color, and replaces the ball, repeating until
he draws the green ball. What is the probability that Dwight will take exactly 4 tries to draw the green
ball from the urn? Use Excel to find the probability, rounding to three decimal places. - Answers Use
Excel to find the probability. Note that this is not a cumulative probability. Let a success be drawing the
green ball.
1. First press FORMULAS and then INSERT FUNCTION.
2. Then select the NEGBINOM.DIST function.
3. Next enter the values for the number of failures before the first success, the number of successes,
and the probability of a success. In this case, since we want to determine the probability of exactly 4
trials, there are 3 failures before the first success. Thus, enter 3, 1, and 1/4, in that order. In the entry for
a cumulative probability, enter 0 since this is not a cumulative probability.
4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.105.
An urn contains 50 total balls, which comprise 49 white balls and one green ball. Dwight is running an
experiment where he draws a ball from the urn, records the color, and replaces the ball, repeating until
he draws the green ball. What is the probability that Dwight will take at most 10 tries to draw the green
ball from the urn? Use Excel to find the probability, rounding to three decimal places. - Answers Use
Excel to find the probability. Note that this is a cumulative probability. Let a success be drawing the
green ball.
,1. First press FORMULAS and then INSERT FUNCTION.
2. Then select the NEGBINOM.DIST function.
3. Next enter the values for the number of failures before the first success, the number of successes,
and the probability of a success. In this case, since we want to determine the probability of at most 10
trials, there are at most 9 failures before the first success. Thus, enter 9, 1, and 1/50, in that order. In the
entry for a cumulative probability, enter 1 since this is a cumulative probability.
4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.182927, which
is 0.183 rounded to three decimal places.
In a recent baseball season, Ron was hit by pitches 21 times in 602 plate appearances during the regular
season. Assume that the probability that Ron gets hit by a pitch is the same in the playoffs as it is during
the regular season. In the first playoff series, Ron has 23 plate appearances. What is the probability that
Ron will get hit by a pitch exactly once? Use Excel to find the probability.
Round your answer to three decimal places. - Answers $0.367$0.367
Note that this is a binomial probability. In this case, we want to find the probability of 1 success, where a
success is getting hit by a pitch. To determine the probability from a binomial distribution using Excel,
follow the steps below.
1. First press FORMULAS and then INSERT FUNCTION.
2. Then select the BINOM.DIST function.
3. Next enter the values for the number of successes, the number of trials, the probability of a success,
and the number of successes. In this case, enter 1, 23, and 21/602, in that order. Enter 0 for Cumulative
since this is not a cumulative probability.
4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.367369, which
is 0.367 rounded to three decimal places.
A state lottery sells instant-lottery scratch tickets. 12% of the tickets have prizes. Neil goes to the store
and buys 10 tickets. What is the probability that exactly three of Neil's tickets will have prizes? Use Excel
to find the probability.
Round your answer to three decimal places. - Answers $0.085$0.085
Note that this is a binomial probability. In this case, we want to find the probability of 3 successes,
where a success is a ticket having a prize. To determine the probability from a binomial distribution
using Excel, follow the steps below.
1. First press FORMULAS and then INSERT FUNCTION.
2. Then select the BINOM.DIST function.
, 3. Next enter the values for the number of successes, the number of trials, the probability of a success,
and the number of successes. In this case, enter 3, 10, and 0.12, in that order. Enter 0 for Cumulative
since this is not a cumulative probability.
4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.084743, which
is 0.085 rounded to three decimal places.
Connor is a statistics student interested in the number of games won by each team per season during
the past 12 years for a certain professional baseball league. He records the total number of wins x for
each team each season and the probability of each value P(x), as shown in the table provided. Use Excel
to find the mean and the standard deviation of the probability distribution.
Round the mean and standard deviation to three decimal places. - Answers $\mu=69.928,\ \
sigma=6.965$μ=69.928, σ=6.965
Step 1: Enter the values of the random variable X in column A and the corresponding probabilities in
column B. Then find the product of column A and column B and enter the results in column C.
Step 2: Add the elements in column C to find the mean μ, which is 69.928.
Step 3: To find the standard deviation, add column D to calculate the product of the square of column A
and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 4,938.434.
Step 4: Calculate the variance by subtracting ∑x2P(x) and the square of the mean. The variance is
approximately 48.509.
Step 5: Find the square root of the variance to find the standard deviation. The standard deviation,
rounded to three decimal places, is σ≈6.965.
Jenna is a meteorologist who is interested in the distribution of the number of days where the daily
maximum temperature was below 32∘F for the 205 major weather stations across the region for the
month of January. She finds the information and records the number of days a weather station had a
maximum temperature below 32∘F, x, and the probability of each value P(x), as shown in the table
provided. Find the mean and the standard deviation of the probability distribution using Excel.
Round the mean and standard deviation to three decimal places. - Answers Step 1: Enter the values of
the random variable X in column A and the corresponding probabilities in column B. Then find the
product of column A and column B and enter the results in column C.
Step 2: Add the elements in column C to find the mean μ, which is 15.464.
Step 3: To find the standard deviation, add column D to calculate the product of the square of column A
and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 292.518.
Step 4: Calculate the variance by subtracting ∑x2P(x) and the square of the mean. The variance is
approximately 53.383.
