Question

Round probabilities to four places. For example, 0.1234. Show your work. For example, if you are adding two probabilities you could do this: 7/59 + 9/59 = 16/59 = 0.2712.

1. You draw one card from a deck of 52 cards. If you get a heart, you win $18. If you get anything else, you pay $5. Note that there are 13 hearts in the deck.

What is the probability of winning the game?

What is the expected value of the game?

If you play the game 100 times, what is your expected gain or loss?

2. A class has sophomores and juniors, and biology and history majors. The table below shows the number of students in each category. There are 59 students in the class. (There are no double-majors.)

a) If one student is chosen at random, what is the probability of selecting a biology major?

b) If one student is chosen at random, what is the probability of selecting a biology major or a sophomore?

c) If two students are chosen at random, what is the probability of selecting a junior history major and then a sophomore biology major? The selections are made without replacement.

d) If one student is chosen at random, what is the probability of selecting a junior, given that the student chosen is a biology major?

e) If two students are chosen at random, what is the probability of selecting a sophomore history major and then another sophomore history major? The selections are made without replacement.

3. Find the following probabilities using the standard normal table:

a) P(z < 1.67)

b) P(z > -1.44)

c) P(-0.87 < z < 1.95)

4. A basketball player has made 65 of 79 free-throws attempted this season. What is the probability that the player will make the next free-throw they attempt?

What is the probability that the player makes their next 10 free-throws? Assume each free-throw is an independent event.

5. Horses in a stable have a mean weight of 950 pounds with a standard deviation of 77 pounds. Weights of horses follow the normal distribution. One horse is selected at random.

a) What is the probability that the horse weighs less than 900 pounds?

b) What is the probability that the horse weigh more than 1,100 pounds?

c) What is the probability that the horse weighs between 900 and 1,100 pounds?

d) What weight is the 90th percentile? (Round to the nearest pound)

6. Your aunt comes to you with an investment opportunity. She needs you to invest $500,000. If the investment succeeds, it will pay you back your investment plus an additional $2 million. If it fails, you get nothing back. She thinks there is a 30% chance of success.

What is the expected value of the investment?

What is the most likely outcome?

  

Answer 1

Answer:

3a)

Given,

P(z < 1.67) = 0.9525403 [since from the standard normal table]

= 0.9525

b)

P(z > -1.44) = 0.9250663 [since from the standard normal table]

= 0.9251

c)

P(-0.87 < z < 1.95) = P(z < 1.95) - P(z < - 0.87)

= 0.9744119 - 0.1921502 [since from the standard normal table]

= 0.7823

5a)

Given,

Mean = 950

Standard deviation = 77

P(X < 900) = P((x-u)/s < (900 - 950)/77)

= P(z < -0.65)

= 0.2578461 [since from the standard normal table]

= 0.2578

b)

P(X > 1100) = P((x-u)/s > (1100 - 950)/77)

= P(z > 1.95)

= 0.0255881 [since from the standard normal table]

= 0.0256

c)

P(900 < X < 1100) = P((900 - 950)/77 < (x-u)/s < (1100 - 950)/77)

= P(-0.65 < z < 1.95)

= P(z < 1.95) - P(z < -0.65)

= 0.9744119 - 0.2578461 [since from the standard normal table]

= 0.7166

d)

Here at 90th percentile, z value is 1.282 [since from the standard normal table]

consider,

X = mean + z*standard deviation

substitute values

= 950 + 1.282*77

= 950 + 98.714

= 1048.714

Please post the remaining questions as separate posts. Thank you.