The CDC starts decides to take 45% of the population of the United States and use the vaccine on them to create an immunity to the virus. Of the people who take the vaccine, 82% receive an immunity to the virus. Of those who do not receive the vaccine, about 7% receive a natural immunity. Use this information to determine the following: 19) What is the probability that a person will develop an immunity to the virus 20) What is the probability that a person will get the vaccine and develop the immunity? 21) What is the probability that a person will get an immunity given they took the vaccine?
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
The CDC starts decides to take 45% of the population of the United States and use the vaccine on them to create an immunity to the virus. Of the people who take the vaccine, 82% receive an immunity to the virus. Of those who do not receive the vaccine, about 7% receive a natural immunity. Use this information to determine the following:
19) What is the
20) What is the probability that a person will get the vaccine and develop the immunity?
21) What is the probability that a person will get an immunity given they took the vaccine?
19)
Consider the event V which is defined the population of Country US who got vaccine.
It is given that P(V) = 0.45.
Consider another event V’, which is defined the remaining population of Country US who did not get vaccine.
Hence, P(V’) = 0.55.
Consider another event I which defines the population who developed an immunity.
It is given that, P(I|V) = 0.82.
It is given that, who do not receive the vaccine, among them 7% receive a natural immunity.
It implies that, P(I|V’) = 0.07.
Hence, the probability that a person will develop an immunity to the virus is,
P(I) = P(I ⋂ V) + P(I ⋂ V’)
= [P(I|V) × P(V)] + [P(I|V’) × P(V’)]
= (0.82 × 0.45) + (0.07 × 0.55)
= 0.4075.
Thus, the probability that a person will develop an immunity to the virus is 0.4075.
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