Sam and Allen have applied for scholarships. The probability that Sam gets a sch

tapetivk 2021-11-20 Answered
Sam and Allen have applied for scholarships. The probability that Sam gets a scholarship is 0.93. The probability that Allen gets a scholarship is 0.77. The probability they both get scholarships is 0.74.
(a) What is the probability that Sam gets a scholarship but Allen doesn't?
(b) What is the probability neither receives a scholarship?
(c) If Sam receives a scholarship, what then is the conditional probability that Allen will also receive a scholarship?
(d) If Allen receives a scholarship, what then is the conditional probability that Sam will not receive a scholarship?

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Expert Answer

Ceitheart
Answered 2021-11-21 Author has 839 answers
Given data:
The probability that Sam got a scholarship is \(\displaystyle{P}{\left({s}\right)}={0.93}\).
The probability that Allen got a scholarship is \(\displaystyle{P}{\left({a}\right)}={0.77}\).
The probability that both got a scholarship is \(\displaystyle{P}{\left({s}\cap{a}\right)}={0.74}\),
(a)
Write the expression for the probability that Sam got a scholarship but Allen doesn't.
\(\displaystyle{p}{\left({A}\right)}={P}{\left({s}\right)}-{P}{\left({s}\cap{a}\right)}\),
Substitute the given values in the above expression.
\(\displaystyle{p}{\left({A}\right)}={0.93}-{0.74}\),
\(\displaystyle={0.19}\)
Thus, there is a 0.19 probability that Sam got a scholarship but Allen doesn't.
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Todd Williams
Answered 2021-11-22 Author has 1224 answers
(b)
Write the expression for the probability that none of them got a scholarship.
\(\displaystyle{p}{\left({B}\right)}={1}-{\left[{P}{\left({s}\right)}+{P}{\left({a}\right)}-{P}{\left({s}\cap{n}\right)}\right]}\)
Substitute the given values in the above expression.
\(\displaystyle{p}{\left({B}\right)}={1}-{\left[{0.93}+{0.77}-{0.74}\right]}\)
\(\displaystyle={0.04}\)
Thus, the probability that that none of them got a scholarship is 0.04.
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user_27qwe
Answered 2021-11-25 Author has 1923 answers

(c)
Write the expression for the conditional probability that Allen also got a scholarship given that Sam got a
scholarship.
\(p\left(\frac{s}{a}\right)=\frac{P\left(s\cap a\right)}{P\left(a\right)}\)
Substitute the given values in the above expression,
\(p\left(\frac{s}{a}\right)=\frac{0.74}{0.93}\)
\(=0.7957\)
Thus, the conditional probability that Allen also got a scholarship given that Sam got a scholarship is \(0.7957.\)
(d)
Write the expression for the conditional probability that Sam will not get a scholarship given that Allen got a
scholarship.
\(p\left(\frac{s`}{a}\right)=\frac{P\left(s`\cap a\right)}{P\left(a\right)}\)
Substitute the given values in the above expression.
\(p\left(\frac{s`}{a}\right)=\frac{0.77-0.74}{0.77}\)
\(=0.03896\)
Thus, the conditional probability that Sam will not get a scholarship given that Allen got a scholarship is
\(0.03896.\)

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