Answer:
A. 0.2734 = 27.34% probability that there are an equal number of boys and girls
B. 0.2753 = 27.53% probability that she’ll answer exactly 5 questions correctly
Step-by-step explanation:
In each of these questions, the binomial probability distribution is used.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinations of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And p is the probability of X happening.
A.The Murray family has produced 8 children.
What is the probability that there are an equal number of boys and girls?
8 children means that ![n = 8](https://tex.z-dn.net/?f=n%20%3D%208)
Each children is equally as likely to be a boy or a girl, which means that ![p = 0.5](https://tex.z-dn.net/?f=p%20%3D%200.5)
This probability is P(X = 4). So
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 4) = C_{8,4}.(0.5)^{4}.(0.5)^{4} = 0.2734](https://tex.z-dn.net/?f=P%28X%20%3D%204%29%20%3D%20C_%7B8%2C4%7D.%280.5%29%5E%7B4%7D.%280.5%29%5E%7B4%7D%20%3D%200.2734)
0.2734 = 27.34% probability that there are an equal number of boys and girls
B.Margie has the uncanny ability of being able to guess the correct answer to TRUE/FALSE questions with an 80% accuracy rate. Tomorrow’s test has 7 questions on it and she needs to answer 5 questions correctly in order to make the Honor Roll. If she guesses on every question, what is the probability that she’ll answer exactly 5 questions correctly?
80% accurracy rate means that ![p = 0.8](https://tex.z-dn.net/?f=p%20%3D%200.8)
7 questions means that ![n = 7](https://tex.z-dn.net/?f=n%20%3D%207)
The probability is P(X = 5). So
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 5) = C_{7,5}.(0.8)^{5}.(0.2)^{2} = 0.2753](https://tex.z-dn.net/?f=P%28X%20%3D%205%29%20%3D%20C_%7B7%2C5%7D.%280.8%29%5E%7B5%7D.%280.2%29%5E%7B2%7D%20%3D%200.2753)
0.2753 = 27.53% probability that she’ll answer exactly 5 questions correctly