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بحل هالمسائل ضروري 
لمادة مبادئ احصاء
Problem 3:
An investment firm has classified its clients according to their gender and the composition of their investment portfolio (bonds, stocks, or a balanced mix of bonds and stocks). The proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender Bonds Stocks Balanced
Male .18 .20 .25
Female .12 .10 .15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Find the following probabilities:
a) The client selected is male
b) The client selected is either male or has a balanced portfolio
c) The client selected is a female and does not have a balanced portfolio
d) The client is male if he has a balanced portfolio
e) The client has not a balanced portfolio if she is a female.
f) Are the events “client is male” and” client has a balanced portfolio” independent?
a) Are the events “client is male” and” client does not have a balanced portfolio” mutually exclusive?
Problem 4:
A fair coin is tossed 3 times. Let x be the number of heads observed in three tosses of the coin.
a) What is the sample space (S) of this experiment (use the Tree Diagram?)
b) Find the probability distribution of X.
c) Find the probability that less than 2 heads are observed.
d) Find the probability that at least 1 head is observed.
e) Find the expected number of heads that will be observed.
f) Find the standard deviation of the number of heads that will be observed.
Problem 5
6.90 Let X represent the number of children in an Egyptian household. The probability distribution of X is as follows:
x 1 2 3 4 5
p(x) .25 .33 .17 .15 .10
What is the probability that a randomly selected Egyptian household will have
a. more than 3 children?
b. between 3 and 5 children (3 and 5 inclusive)?
c. fewer than 4 children?
d. Find the expected number of children in a randomly selected Egyptian household.
e. Find the standard deviation of the number of children in an Egyptian household.
Problem 5
In a typical month, an insurance agent presents life insurance plans to 40 customers. Historically, 1 in 4 customers choose to buy life insurance from this agent.
a) What is the probability that exactly 5 customers will buy life insurance from this agent?
b) What is the probability that more than 10 customers will buy life insurance from this agent?
c) What is the probability that less than 20 customers will buy life insurance from this agent?
d) Calculate the mean number of customers who will buy a life insurance from this agent?
e) Calculate the standard deviation of the number of customers who buy a life insurance from this agent in a month?