Question

1. Theater tickets for a hit show have four prices depending on seating. The prices ae $50, $100, $150 and $200. The probability a ticket sells for $50 is .4. The probability it sells for $100 is .15. The probability it sells for $150 is .2.

1. Find the probability a ticket sells for $200.

2. Find the expected cost (mean cost) of a ticket.

3. Find the standard deviation for the cost of a ticket

4. Find the variance for the cost of a ticket

2. A person pays $2.00 to play the following carnival game: A die is rolled one time. If an even number comes up the person pays an additional $2.50. If it comes up odd, the player receives a payment of $5.00 and gets the $2.00 back. Find the players expected profit if (s)he plays this game.

3. Sixty percent of all shoppers in a given shopping center use credit cards for their purchases. If 20 shoppers make purchases, find the probability that:

1. exactly 12 use credit cards.

2. exactly 7 do not use credit cards.

3. At most 10 use credit cards

4. More than 13 use credit cards

4. The probability a person passes the Bar exam is .46. If 290 people in this city take the exam:

1. Find the mean number who pass.

2. Find the standard deviation for the number who pass.

5. Find each of the following probabilities for a value chosen at random from a Standard Normal (Z) distribution.

1. the probability the value is more than 2.7

2. the probability the value is between –2.01 and 2.21

3. The probability the value is less than –3.2

6. The hourly wage for workers in a fast food restaurant is Normally distributed with a mean of $5.85 and a standard deviation of $0.35. If a worker is selected at random, find the probability that:

1. (S)he earns less than $5.50 an hour

2. (S)he earns between $5.90 and $6.40 an hour.

3. (S)he earns at least $6.00 an hour

7. The mean cost of living for a family of four in cities across the country is $65,351 with a standard deviation of $7712. A company is thinking of relocating to a city with a cost of living that is in the bottom 40% of all the cities. Assuming that the distribution is Normally distributed, what is the cutoff score for a city that would make it eligible for consideration by this company?

8. 45% of people have type O blood . If 400 volunteers show up to donate blo, use the Normal approximation to the binomial to find the probability that more than 175 but at most 182 have type O blood.

9. The mean annual rainfall in a particular region is 80 inches with a standard deviation of 8 inches. If a sample of 32 years is selected, what is the probability that the mean annual rainfall for this sample will be less than 79 inches?

Answer 1

1)

a)

Using the law of total probability

b)

The expected value is obtained using the following formula,

From the data values,

Prices, X	Probability, P(X)	X*P(X)
50	0.4	20
100	0.15	15
150	0.2	30
200	0.25	50
.	Sum	115

c)

The standard deviation value is obtained using the following formula,

From the data values,

Prices, X	Probability, P(X)	X*P(X)	X^2*P(X)
50	0.4	20	1000
100	0.15	15	1500
150	0.2	30	4500
200	0.25	50	10000
.	Sum	115	17000

d)

The variance value is obtained using the following formula,

2)

If the die is fair, the probability of getting even number = 0.5 and the probability of getting odd number = 0.5

If even number comes, the player's profit = -(2 + 2.5) = - $4.5

If odd number comes, the player's profit  = 2 + 5 = $7

The expected profit value is obtained using the following formula,

From the data values,

	X	P(X)	X*P(X)
Even number	-4.5	0.5	-2.25
Odd number	7	0.5	3.5
			1.25

3)

Let the random variable, X = number of shoppers who uses credit cards for their purchases in a given shopping center,

When we want to calculate the probability of the number of success in a fixed number of trails (Bernoulli's trials i.e. two possible outcomes, success/failure), the binomial distribution follows.

Hence the sample size of 20 shoppers will have a binomial distribution with parameters n and p.

n = number of trials = 20

p = probability of success = 0.60

a)

exactly 12 use credit cards.

The probability mass function for the binomial distribution is defined as,

Now, the required probability is,

(In excel use function =BINOM.DIST(12,20,0.6,FALSE))

b)

exactly 7 do not use credit cards.

Now the probability of success (don not use credit cards) = 1 - 0.6 = 0.4

Now, the required probability is,

(In excel use function =BINOM.DIST(7,20,0.4,FALSE))

c)

At most 10 use credit cards

The cumulative distribution function is defined as,

Now, the required probability is,

(In excel use function =BINOM.DIST(10,20,0.6,TRUE))

d)

More than 13 use credit cards

Now, the required probability is,

(In excel use function =1-BINOM.DIST(13,20,0.6,TRUE))

4)

Let the random variable, X = number of persons passes the Bar exam

When we want to calculate the probability of the number of success in a fixed number of trails (Bernoulli's trials i.e. two possible outcomes, success/failure), the binomial distribution follows.

Hence the sample size of 290 persons will have a binomial distribution with parameters n and p.

n = number of trials = 290

p = probability of success = 0.46

a)

The mean value is obtained using the following formula,

b)

The standard deviation value is obtained using the following formula,