MA 3210, Midterm 1, Type I ,Spring 2000

Show all your work. NAME:

	1	2	3	4	5	6	7	8	9	10	Total

1.

(a)

A and B are mutually exclusive events. P(A)=0.3, P(B)=0.4. Find $P(A \cup B)$ if you have enough information.

(b)

Let A: the event one likes an apple, B: the event one likes banana. Adam finds that P(A)=0.5, P(B)=0.3. He finds that among the people who like apple, 40 % like apple. Find the probability that one likes both apple and banana.

(c)

Is the event A and B in (2) independent? Justify your answer.

(d)

P(A|E)=0.5. Find $P(A^{\prime} \vert E)$ if there is enough information.

(e)

P(A|E)=0.5. Find $P(A \vert E^{\prime})$ if there is enough information.

(f)

Three sets A, B, C form a partition of S. P(A)=0.2, P(B)=0.4, P(E)=0.5. What is P(C)?

(g)

P(A)=0.5, P(E)=0.3 and P(A|E)=0.6.
Find $P(A\cap E), P(A \cap E^{\prime}), P(A \vert E^{\prime})$ for him if he has enough information.

(h)

P(A)=0.5, P(B)=0.3 and A, B are independent. Find $P(A \cap B)$.

(i)

$P(E)=0.6, P(A \vert E)=0.3, P(A \vert E^{\prime})=0.5$. Find P(E|A).

2.

The probability that OW students like Nike product is 60% and the probability that OW students like LA Gear product $30 \%$, and the probability that OW students like both Nike and LA Gear is $15 \%$.

(a)

Find the probability that a randomly chosen OW student like Nike or LA Gear

(b)

Find the probability that a randomly chosen OW student like Nike but not LA Gear

(c)

Find the probability that a randomly chosen OW student like Neither Nike nor LA Gear

3.

How many ways are there to distribute 9 scientists into two bedroom, 3 bedroom, four bedroom units.

4.

How many permuations are possible to arrange the letters in "gigantic"?

5.

In a poker hand consisting of 5 cards, find the probability of holding

(a)

4 aces and 1 King.

(b)

spades and 2 club and 1 herat.

6.

In the experiment of tossing a coin 6 times,

(a)

how many different outcomes are possible?

(b)

how many outcomes with 4 heads and 2 tails are possible?

(c)

What is the probability that having 3 heads and 3 tails?

7.

A coin is biased so that head is twice likely to occur as a tail. If a coin is tossed four times, what is the probability of getting 3 head and 1 tail?

8.

A box contains 3 nickes and 2 dime. If a nickel is drawn, Judy draws one ball from Bag 1, and if a dime is drawn then Judy draws a ball from Bag II. Bag I contains 4 red balls and 2 blue balls, Bag II contains 3 red balls and 3 blue balls.

(a)

Complete the probability tree for this experiment.

(b)

What is the probability $P(R\vert N), P(R), P(R \cap N)$, where
R denotes the event of drawing a red ball, B denotes the event of drawing a blue ball and N and D denote the event of drawing a nickel and a dime respectively.

(c)

Describe P(N|R), P(D|R) in words and find them.

9.

Two fair dice are tossed. Adam is interested in finding the probability of sum of the two dice and Judy is interested in finding the minimum number of the two dice.
Part 1:

(a)

Find the probability that the sum is 9 for Adam.

(b)

Find the probability that the sum is 7.

(c)

Find the probability that the min number is 5 for Judy.

Part 2: Adam uses X for his random variable and Judy decided to denote her random variable as Z.

(a)

Find the outcomes (event) corresponding X=9 and P(X=9)

(b)

Find P(X=3)

(c)

Find all outcomes whose random variable Z takes the value 4.
Find P(Z=4).

(d)

Find P(Z=5).

10.

Long Island Railroad has 3 lines. L₁, L₂, L₃ which make up 25%, 35 %, and 40% of LIRR service. The data shows that 3 %, 2% and 4 % of each line are running late.

(a)

If a train is randomly slected, what is the probability that it is running late?

(b)

A train is randomly selected from the ones which run late. What is the probability that it is on line L₂?