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# A Discussion On Probability Distributions

6. A coin is tossed 4 times. Let X denote the number of heads which appear in 4 tosses.
a) Construct a probability distribution for X
b) Find P(X>2)
c) Find the expected value of X
d) Find the variance and standard deviation of X

7. The probability that a randomly selected elementary or secondary school teacher from a city is female is 0.68, holds a second job is 0.38, and is female and holds a second job is 0.29. Find the probability that an elementary or secondary school teacher selected at random from this city is female or holds a second job.

8. Suppose the statistic final exam has 10 multiple choice questions, each with 4 choices. If a student randomly guesses the answers, what is the probability he will pass the final exam. (Assume the student passes with 60 or higher, and each question is worth 10)

#### Solution Preview

6. a.
Probability distribution,
P(X = r) = C(n,r) p^r q^(n-r)

p = 1/2 = 0.5, q = 1/2 = 0.5, n = 4

P(X=r) = C(4,r)*0.5^r * 0.5^(4-r) --Answer

b.
P(X >=2) = 1 - P(X<2) =
= 1 - P(X=0) - P(X=1)
= 1 - C(4,0)*p^0 * q^4 - C(4,1)*p^1 * q^3
= 1 - 1*1*0.5^4 - 4*0.5^1*0.5^3 = 0.6875 ...

#### Solution Summary

Here we solve some problems related to binomial distribution (probability for obtaining different cases, mean, SD) in coin toss and in multiple choice questions, and an example of combined probability.

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