In Class Exam 3 Review Math 141

 

1. A medical test for use of a certain drug is positive in 90% of those who use the drug. The test is also positive in 7% of those who do not use the drug. 12% of the people tested use the drug.

 

 

 

 

 

 

 

 

 

 

 

a) What is the probability someone who does not use the drug gets a negative test?

 

 

b) What is the probability someone who gets a positive test uses the drug?

 

 

c) Assuming successive tests are independent, what is the probability someone who gets two positive tests uses the drug?

 

 

 

2. A person plays the following game. He draws 1 card from a standard 52-card deck. If it’s an ace, he chooses 1 of 2 doors. Behind one door is a diamond, which he can exchange for $3000.

Behind the other door is a piece of coal.

            If the card he draws is not an ace, he keeps the card and chooses again. If the second card matches the first card’s denomination, he chooses 1 of 3 doors. One door has the $3000 diamond, one has $10 and the other has the piece of coal. If the second card does not match the first, he wins nothing.

a) Find his expected winnings.

 

 

 

 

 

 

 

 

b) If he pays $180 to play the game, what is his expected net gain?

 

3. A student takes 12 credits and receives the following grades:

        grade   A    B     C     D

# credits      6     2      3     1

 

He previously had 60 credits with a gpa of 2.8. Find his new gpa.

 

 

4. I.   E and F are two events in a sample space. P(E)=0.7, P(F)=0.6 and P(= 0.1.

a) Are E and F independent?

 

 

b) Find P(E|F).

 

II.    If P(E)=0.75, P(F)=0.8 and P()=0.05 are E and F independent? Find P(E|F).

 

 

 

 

5. The events that runners I, II and III can run a mile in 4 minutes are independent and have probabilities 0.6, 0.9 and 0.7 respectively.

 

a) Find the probability that at least one runner will not run the mile in 4 min.

 

 

b) Find the probability that only runner II will finish in 4 min.

 

c) Find the probability that runners I or II, possibly both, will finish in 4 min.

 

 

6. A die is tossed 3 times.

a) What is the probability that the 2nd toss is a “6” given that there are exactly two “6”s?

 

 

 

 

 

 

b) Are the events that the ‘2nd toss is a “6” ‘and ‘there are exactly two “6”s ‘ independent?

 

 

 

 

c) Answer a and b if the die is tossed 12 times.

 

 

 

 

 

7. Classify each random variable as continuous, infinite discrete or finite.

A coin is randomly selected from a box of coins containing nickels, dimes, quarters and pennies. Then the coin is returned to the box.

a) X is the money value of the coin.

 

b) Y is the un-rounded weight of the person choosing the coin.

 

c) N is the number of times a coin must be selected until a nickel is chosen.

 

8. I A person draws one card from a standard 52-card deck.

a) What are the odds he will select an ace?

 

 

b) What are the odds he will not select an ace?

 

 

II  If the odds in favor of E are 2:9, find P(E).

 

 

9. Classify each random variable as binomial or not. If  binomial, give n, p, the mean and the standard deviation. If not binomial, say why not.

 

a) A shipment contains 100 games, two of which are defective. 10 games are selected at random and X is the number of defective games.

 

 

 

b) Games are continually produced so that 2% are defective. A sample of 10 games are selected at random and X is the number of defective games.

 

 

c) 1/3 of the population has blood type A+. Fifty people are selected at random and X is the number who have blood type A+.

 

 

d) A box contains 10 red, 6 blue and 4 green balls. Three are randomly selected without replacement. X= the number of green balls chosen.

 

 

10. Write the distribution of the random variable in 8d. Find E(X).

 

 

 

 

 

 

11. X is a normally distributed random variable with mean 30 and unknown standard deviation.

P(X>25)=0.8. Find

a) P(X<35)       b) P(X>35)     c) P(25<X<30)

 

 

 

12. A set of grades is normally distributed with mean 70 and standard deviation 16.

i) Find a number a so that 60% of the grades are below a.

 

 

 

ii) Find a number b so that 15% of the grades are above b.

 

iii) Find P(X<80) where X is a randomly selected student’s grade.

13. Assume that 6% of a large population has a non-communicable disease. 500 people are randomly selected and X is the number of those selected who have the disease.

 

a) Find P(20<X<30).

 

 

 

 

b) Find the normal approximation to the probability in a.

 

14. 15 people are in a room. What is the probability that

 

a)      at least two have the same birthday?

b)      exactly two have the same birthday and there are no other repeats?