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# 3-2 Problem Set: Probability

#### A patient newly diagnosed with a serious ailment is told he has a 60% probability of surviving 5 or more years. Let us assume this statement is accurate. Explain the meaning of this statement to someone with no statistical background in terms he or she will understand.

##### The probability of the patient living for 5 or more years is 60%, and the probability of the patient not surviving at least 5 years is 40%. The patient has a better chance of surviving at least 5 years than dying within 5 years.

In order to explain this to someone with no statistical background, we can explain it in following ways:

1. The patients has 60% chance of surviving 5 or more years.
2. The patient has 40% chance of not surviving 5 or more years.
3. The chances of survival are greater but we cannot conclude that the person will surely live for 5 or more years.

We can also explain from a sample containing greater number of patients as discussed below:

60% of patients with this ailment will survive for 5 or more years.

40% of patients will not survive for 5 or more years.

Suppose a population has 26 members identified with the letters A through Z.

1. You select one individual at random from this population. What is the probability of selecting individual A?

# 1/26 or .038

• Assume person A gets selected on an initial draw, you replace person A into the sampling frame, and then take a second random draw. What is the probability of drawing person A on the second draw?

# 1/26 or .038 because once A gets placed back into the frame, it becomes the original set.

• Assume person A gets selected on the initial draw and you sample again without replacement. What is the probability of drawing person G on the second draw? 1/25 or .04
• Let A represent cat ownership and B represent dog ownership. Suppose 35% of households in a population own cats, 30% own dogs, and 15% own both a cat and a dog. Suppose you know that a household owns a cat. What is the probability that it also owns a dog?

P(B)=.3

# P(AB)=.15 P(AB)/P(A)=.15/.35=.43

• What is the complement of an event?

# The complement of an event is the subset of outcomes in the sample space that are not in the event.

• Suppose there were 4,065,014 births in a given year. Of those births, 2,081,287 were boys and 1,983,727 were girls.
1. If we randomly select two women from the population who then become pregnant, what is the probability both children will be boys?

# P(BB)=(2,081,287/4,065,014) x’s (2,081,287/4,065,014)

=.262 or 26.2%

• If we randomly select two women from the population who then become pregnant, what is the probability that at least one child is a boy?

# P(BB) = 0.262 ; P(GB) = 0.5 0.262 + 0.5 = 0.762 or 76.2%

Data for the  problem from the attached file

Registered birth = 4,065,014

Number of boys = 2,081,287

Number of girls  = 1,983,727

Let Pb be the probability for boy to born,

Pb = 2,081,287/4,065,014 = 0.512

Thus, probability for girl to born,

Pg = 1,983,727/4,065,014 = 0.488

Now,

Two women were randomly selected out of which what is probability that at least one child is boy,

Let, two women’s be A1 and A2,

A1              A2             Final probability

Boy           Girl               = Pb*Pg = 0.512*0.488 = 0.249856

Girl            Boy              = Pg*Pb = 0.488*0.512 = 0.249856

Boy           Boy               = Pb*Pb = 0.512*0.512 = 0.262144

Thus,

Total probability is = 0.249856 + 0.249856 + 0.262144

= 0.76185

or

= 0.762 (up-to three decimals)

Thus, total probability is 0.762 or 76.2%, that at least one child is boy.

• Explain the difference between mutually exclusive and independent events.

# The difference between mutually exclusive and independent events is that a mutually exclusive event is a situation where two events cannot occur at same time where an independent event is where one event remains unaffected by the occurrence of the other event.

Mutually exclusive is a statistical term describing two or more events that cannot happen simultaneously. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other while an independent event is an event that has no connection to another event’s chances of happening. In other words, the event has no effect on the probability of another event occurring. When two events are independent, one event does not influence the probability of another event.

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