Respuesta :
Answer:
0.9375 = 93.75% probability that at least one of the four children is a girl.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
We have the following sample space
In which b means boy, g means girl
b - b - b - b
b - b - b - g
b - b - g - b
b - b - g - g
b - g - b - b
b - g - b - g
b - g - g - b
b - g - g - g
g - b - b - b
g - b - b - g
g - b - g - b
g - b - g - g
g - g - b - b
g - g - b - g
g - g - g - b
g - g - g - g
Total outcomes
There are 16 total outcomes(size of the sample space)
Desired outcomes
Of these outcomes, only 1(b - b - b - b) there is not a girl.
So the number of desired outcomes is 15.
Probability:
[tex]P = \frac{15}{16} = 0.9375[/tex]
0.9375 = 93.75% probability that at least one of the four children is a girl.
The probability of an event is how likely the event is to happen. The probability that at least one of the 4 children is a girl is 0.9375.
Given that:
[tex]n = 4[/tex]
[tex]b \to boys \\ g \to girls[/tex]
[tex]b = 0.5\\ g = 0.5[/tex] ----- Because both have equal probabilities
This probability is an illustration of binomial probability where:
[tex]P(x) = ^nC_x g^x b^{n-x}[/tex]
The probability that at least one of the 4 children is a girl is represented as:
[tex]P(x \ge 1)[/tex]
Using complement rule:
[tex]P(x \ge 1) = 1 - P(x = 0)[/tex] ---- i.e. 1 - the probability that none of the children is a girl
So, we have:
[tex]P(x) = ^nC_x b^x g^{n-x}[/tex][tex]P(x) = ^nC_x g^x b^{n-x}[/tex]
[tex]P(x = 0) = ^4C_0 \times 0.5^0 \times 0.5^{4-0}[/tex]
[tex]P(x = 0) = ^4C_0 \times 1 \times 0.5^4[/tex]
[tex]P(x = 0) = ^4C_0 \times 1 \times 0.0625[/tex]
[tex]P(x = 0) = 1 \times 1 \times 0.0625[/tex]
[tex]P(x = 0) = 0.0625[/tex]
So, we have:
[tex]P(x \ge 1) = 1 - P(x = 0)[/tex]
[tex]P(x \ge 1) = 1 - 0.0625[/tex]
[tex]P(x \ge 1) = 0.9375[/tex]
Hence, the probabilities that at least one of the 4 children is a girl is 0.9375.
Read more about probabilities at:
brainly.com/question/11234923
