Two fair coins are tossed. Let E be the event \"the coins match\", and F be the

Question

Two fair coins are tossed. Let E be the event "the coins match", and F be the event "at least one coin is tails". (a) What is the sample space S of this experiment (tossing two coins)? (b) Find the probabilities P(E) and P(F). (c) Find the probabilities P(E union F) and P(E intersection F). (d) Are the events E and F independent? Explain. Consider the word ALABAMA. (a) In how many distinct ways can the letters of this word be rearranged? (b) What is the probability that such a random rearrangement has at least two consecutive As? Let X be a discrete random variable with probability function f(x) = k(x^2 - 2x) + 0.2 for x = 0, 1, 2, 3. (a) Find the value of k. (b) Find f(2.75) and F(2.75). (c) Find the mean mu. (d) Find the variance sigma^2. According to government data, 25% of employed women have never been married. Suppose 192 employed women are selected at random. (a) What is the probability that exactly 48 of them have never been married? (b) Use the normal approximation to find the probability that the number of women who have never been married will be between 40 and 52? Books from a certain publisher contain an average of 2 misprint per page. Use Poisson distribution to find the probability that a given page contains at least 3 misprints.

Explanation / Answer

a)

f(0) + f(1) + f(2) + f(3) = 1

k(0 - 2.0) + 0.2 + k(1 -2) +0.2 + k(4-4) + 0.2 + k(9-6)+0.2 = 1

=>

-k +3k + 0.8 = 1

=> k = 0.1

b)

f(2.75) is not possible since it is discreet

F(2.75) = f(0) + f(1) + f(2)

= 0.2 + 0.1 +0.2

= 0.5

c)

mean = sum of (x * f(x))

= 0 * 0.2 + 1 * 0.1 + 2 * 0.2 + 3 * 0.5

= 2

d)

E(x^2) = sum of (x^2 * f(x))

= 0 * 0.2 + 1 * 0.1 + 2^2 * 0.2 + 3^3 * 0.5

= 5.4

variance = 5.4 - 2^2

= 1.4

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