Question

‼️PLS HELP ‼️

1) As a special promotion for its 20-ounce bottles of soda,
a soft drink company printed a message on the inside of
each cap. Some of the caps said "Please try again,"
while others said, "You're a winner!" The company
advertised the promotion with the slogan "1 in 6 wins a
prize" Suppose the company is telling the truth and that
every 20-ounce bottle of soda it fills has a 1-in-6 chance
of being a winner. Seven friends each buy one 20-ounce
bottle of the soda at a local convenience store.
Is this binomial or geometric? If X=the number who win a prize, find P(X>3). Round to the nearest thousandth.

2) As a special promotion for its 20-ounce bottles of soda,
a soft drink company printed a message on the inside of
each cap. Some of the caps said "Please try again,"
while others said, "You're a winner!" The company
advertised the promotion with the slogan "1 in 6 wins a
prize" Suppose the company is telling the truth and that
every 20-ounce bottle of soda it fills has a 1-in-6 chance
of being a winner. Alan decides to keep buying one 20-
ounce bottles of soda at a time until he gets a winner.
Is this binomial or geometric? What is the probability he buys exactly 5 bottles of
soda? Round to the nearest hundredth.

3) According to a study published by a group of University
of Massachusetts sociologists, about two-thirds of the
20 million persons in this country who take Valium are
women. If a random sample of 38 Valium users is taken,
what is the probability that 20 of them are women?
Is this binomial or geometric? What is the probability that 20 of them are
women? Round to the nearest thousandth.

4) Seed Depot advertises that its new flower seeds have an
85% chance of germinating (growing). Suppose that the
company's claim is true. Judy gets a packet with 20
randomly selected new flower seeds from Seed Depot
and plants them in her garden. What is the probability
that less than 12 germinate?
Is this binomial or geometric? What is the probability that less than 12
germinate? Round to 4 decimal places.

5) Thousands of travelers pass through the airport in
Guadalajara, Mexico, each day. Before leaving the
airport, each passenger must go through the customs
inspection area. Customs agents want to be sure that
passengers do not bring illegal items into the country.
But they do not have time to search every traveler's
luggage. Instead, they require each person to press a
button. Either a red or a green bulb lights up. If the red
light flashes, the passenger will be searched by customs
agents. A green light means "go ahead." Customs
agents claim that the light has probability 0.30 of
showing red on any push of the button. Assume for now
that this claim is true. Suppose we watch 20 passengers
press the button. Let R = the number who get a red light.
What is the mean of R? Interpret this value. What is the standard deviation of R? Interpret this
value.

6) Thousands of travelers pass through the airport in
Guadalajara, Mexico, each day. Before leaving the
airport, each passenger must go through the customs
inspection area. Customs agents want to be sure that
passengers do not bring illegal items into the country.
But they do not have time to search every traveler's
luggage. Instead, they require each person to press a
button. Either a red or a green bulb lights up. If the red
light flashes, the passenger will be searched by customs
agents. A green light means "go ahead." Customs
agents claim that the light has probability 0.30 of
showing red on any push of the button. Assume for now
that this claim is true. Suppose we watch 20 passengers
press the button. Let R= the number who get a red light.
What is the mean of R? Interpret this
value. What is the standard deviation of R? Interpret this
value.

Answer 1

The scenarios described involve various probability distributions, including binomial and geometric. Step-by-step explanations are provided for finding probabilities, means, and standard deviations for each scenario.

Step-by-step explanation:

1) Binomial Distribution:

The scenario with the friends buying the soda bottles can be modeled using the binomial distribution.

For X = number of friends who win a prize, we are asked to find P(X > 3).

In this case, we have 7 friends, each with a 1/6 chance of winning a prize. We can use the binomial probability formula to calculate P(X > 3).

P(X > 3) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

2) Geometric Distribution:

In the second scenario with Alan buying sodas until he gets a winner, we have a geometric distribution.

We want to find the probability that Alan buys exactly 5 bottles of soda.

We can use the geometric probability formula:

P(X = 5) = (1-p)^(x-1) * p

3) Binomial Distribution:

In the Valium study scenario, we have a binomial distribution.

We want to find the probability that 20 out of 38 Valium users are women.

We can use the binomial probability formula:

P(X = 20) = nCk * p^k * (1-p)^(n-k)

4) Binomial Distribution:

In the Seed Depot scenario, we also have a binomial distribution.

We want to find the probability that less than 12 seeds germinate.

We can use the binomial probability formula to calculate P(X < 12).

5) Binomial Distribution:

In the airport customs scenario, we have another binomial distribution.

We want to find the mean and standard deviation of the number of passengers who get a red light.

The mean of a binomial distribution is given by μ = np, and the standard deviation is given by σ = sqrt(npq).

6) Binomial Distribution:

In the second airport customs scenario, we have a binomial distribution.

We want to find the mean and standard deviation of the number of passengers who get a red light.

The mean of a binomial distribution is given by μ = np, and the standard deviation is given by σ = sqrt(npq).