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A couple intends to have two children, and suppose that approximately 52% of births are male and 48% are female.
a. What is the probability both children are male?
b. What is the probability both children are female?
c. What is the probability they have exactly one male and exactly one female child?

Respuesta :

skyluke89

a) Probability of both being males is 27%

b) Probability of both being females is 23%

c) Probability of having exactly one male and one female is 50%

Step-by-step explanation:

a)

The probability that the birth is a male can be written as

[tex]p(m) = 0.52[/tex] (which corresponds to 52%)

While the probability that the birth is a female can be written as

[tex]p(f) = 0.48[/tex] (which corresponds to 48%)

Here we want to calculate the probability that over  2 births, both are male. Since the two births are two independent events (the probability of the 2nd to be a male  does not depend on the fact that the 1st one is a male), then the probability of both being males is given by the product of the individual probabilities:

[tex]p(mm)=p(m)\cdot p(m)[/tex]

And substituting, we find

[tex]p(mm)=0.52\cdot 0.52 = 0.27[/tex]

So, 27%.

b)

In this case, we want to find the probability that both children are female, so the probability

[tex]p(ff)[/tex]

As in the previous case, the probability of the 2nd child to be a female is independent from whether the 1st one is a male or a female: therefore, we can apply the rule for independent events, and this means that the probability that both children are females is the product of the individual probability of a child being a female:

[tex]p(ff)=p(f)\cdot p(f)[/tex]

And substituting

[tex]p(f)=0.48[/tex]

We find:

[tex]p(ff)=0.48\cdot 0.48=0.23[/tex]

Which means 23%.

c)

In this case, we want to find the probability they have exactly one male and exactly one female child. This is given by the sum of two probabilities:

- The probability that 1st child is a male and 2nd child is a female, namely [tex]p(mf)[/tex]

- The probability that 1st child is a female and 2nd child is a male, namely [tex]p(fm)[/tex]

So, this probability is

[tex]p(mf Ufm)=p(mf)+p(fm)[/tex]

We have:

[tex]p(mf)=p(m)\cdot p(f)=0.52\cdot 0.48=0.25[/tex]

[tex]p(fm)=p(f)\cdot p(m)=0.48\cdot 0.52=0.25[/tex]

Therefore, this probability is

[tex]p(mfUfm)=0.25+0.25=0.50[/tex]

So, 50%.

Learn more about probabilities:

brainly.com/question/5751004

brainly.com/question/6649771

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