# Thread: Probability of acceptance

1. ## Probability of acceptance

There are two submissions for a university. A and B.
The applicant feels that there is a 60% chance of receiving an offer from university A, and a 50% chance of receiving one from university B.
If the applicant receives an offer from university B, then the applicant will feel that there is an 80% chance of receiving an offer from university A.

What are the chances of receiving an offer from university A.
What are the chances of not receiving an offer from A.

What are the chances that both universities will make an offer.
What is the probability that at least one university will make an offer.

If an offer from university B is made, what is the probability that there will be no offer from university A.

2. Hello, NumbersDontLie!

There are submissions for two universities: $A$ and $B.$

There is a 60% chance of receiving an offer from university $A$, .[1]
and a 50% chance of receiving one from university $B$. .[2]

If the applicant receives an offer from university $B$,
then there is an 80% chance of receiving an offer from university $A$. .[3]

(a) What is the probability of receiving an offer from university $A$?
(b) What is the probability of not receiving an offer from $A$?
(c) What is the probability that both universities will make an offer?
(d) What is the probability that at least one university will make an offer?
(e) If there is an offer from university $B$, what is the probability
. . .that there will be no offer from university $A$?

From [1], we have: . $P(A) \:=\:0.6,\quad P(^{\sim} A) \:=\:0.4$

From [2], we have: . $P(B) \:=\:0.5,\quad P(^{\sim} B) \:=\:0.5$

From [3], we have: . $P(A\,|\,B) \:=\:0.8 \quad\Rightarrow\quad \frac{P(A \wedge B)}{P(B)} \:=\:0.8$

. . $P(A\wedge B) \:=\:(0.8)\!\cdot\! P(B) \:=\:(0.8)(0.5) \quad\Rightarrow\quad P(A \wedge B)\:=\:0.4$

We have enough information to complete the probability chart.

. . . . $\begin{array}{|c||c|c||c|} \hline
& B & ^{\sim} B & \text{Total} \\ \hline \hline
A & 0.4 & 0.2 & 0.6 \\ \hline
^{\sim} A & 0.1 & 0.3 & 0.4 \\ \hline \hline
\text{Total} & 0.5 & 0.5 & 1.0 \\ \hline \end{array}$

which provides the information needed to answer the questions.