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

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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Ceitheart
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
(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.
user_27qwe

(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
$$0.03896.$$