Chapter 13, Problem 12CT

Kiersten applies for admission to the University of Southern California (USC) and Florida State University (FSU). She estimates that she has a $60\%$ chance of being admitted to USC, a $70\%$ chance of being admitted to FSU, and a $35\%$ chance of being admitted to both universities.

What is the probability that she will be admitted to either USC or FSU?

What is the probability that she will not be admitted to FSU?

To determine

To calculate: The probability that Kiersten will be admitted to either USC or FSU, if Kiersten estimates that she has a $60\%$ chance of being admitted to USC, a $70\%$ chance of being admitted to FSU and a $35\%$ chance of being admitted to both universities.

Answer to Problem 12CT

Solution:

The probability that Kiersten will be admitted to either USC or FSUis $0.95$.

Explanation of Solution

Given information:

Kiersten estimates that she has a $60\%$ chance of being admitted to USC, a $70\%$ chance of being admitted to FSU, and a $35\%$ chance of being admitted to both universities.

Formula used:

Probability of the union of two events is;

$P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)$.

Calculation:

Denote, the event that Kiersten of being admitted to USC by $E$ and the event that Kiersten of being admitted to FSU by $F$.

Kiersten estimates that she has a $60\%$ chance of being admitted to USC, a $70\%$ chance of being admitted to FSU, and a $35\%$ chance of being admitted to both universities.

Thus, $P\left(E\right)=0.60$, $P\left(F\right)=0.70$ and $P\left(E\cap F\right)=0.35$.

To find the probability that Kiersten will be admitted to either USC or FSU, that to find $P\left(E\cup F\right)$, use the formula for probability of the union of two events.

Thus, $P\left(E\cup F\right)=0.60+0.70-0.35=0.95$.

Therefore, the probability that Kiersten will be admitted to either USC or FSU is $0.95$.

To determine

To calculate: The probability that Kiersten will not be admitted to FSU, if Kiersten estimates that she has a $60\%$ chance of being admitted to USC, a $70\%$ chance of being admitted to FSU, and a $35\%$ chance of being admitted to both universities.

Answer to Problem 12CT

Solution:

The probability that Kiersten will not be admitted to FSU is $0.30$.

Explanation of Solution

Given information:

Kiersten estimates that she has a $60\%$ chance of being admitted to USC, a $70\%$ chance of being admitted to FSU, and a $35\%$ chance of being admitted to both universities.

Formula used:

The formula to find probability of the complement of an event $E$ is;

$P\left(\overline{E}\right)=1-P\left(E\right)$.

Calculation:

Denote, the event of Kiersten being admitted to USC by $E$ and the event of Kiersten being admitted to FSU by $F$.

Kiersten estimate that she has a $70\%$ chance of being admitted to FSU, and the event Kiersten will not be admitted to FSU is compliment of an event $F$.

Therefore, use the formula to find probability of the complement of an event.

Thus, $P\left(\overline{F}\right)=1-P\left(F\right)$.

$\Rightarrow P\left(\overline{F}\right)=1-0.70=0.30$.

Therefore, the probability that Kiersten will not be admitted to FSU is $0.30$.