Two professors are applying for grants. ProfessorJane has a probability of 0.62 of being funded. Professor Joe hasprobability 0.25 of being funded. Since the grants are submitted totwo different federal agencies, assume the outcomes for each grantare independent. 1.What is the probability that both professors get their grantsfunded? Give your answer to four decimal places. 2.What is the probability that at least one of the professors will befunded? Give your answer to four decimal places. 3.What is the probability that Professor Jane is funded but ProfessorJoe is not? Give your answer to four decimal places. 4.Given at least one of the professors is funded, what is theprobability that Professor Jane is funded but Professor Joe is not?Give your answer to four decimal places.

1. To calculate the probability that both professors get their grants funded, we will use the fact that the events are independent. The probability that both events happen is the product of their individual probabilities. Therefore, the probability that both Professor Jane and Professor Joe get funded is:

P(Jane gets funded and Joe gets funded) = P(Jane gets funded) * P(Joe gets funded) = 0.62 * 0.25 = 0.155

Rounded to four decimal places, the probability is 0.1550.

2. The probability that at least one of the professors will be funded is equivalent to one minus the probability that neither gets funded. To find the probability of neither getting funded, we calculate the product of their probabilities of not getting funded:

P(neither gets funded) = (1 - P(Jane gets funded)) * (1 - P(Joe gets funded)) = (1 - 0.62) * (1 - 0.25) = 0.38 * 0.75 = 0.285

Now, to find the probability of at least one getting funded:

P(at least one gets funded) = 1 - P(neither gets funded) = 1 - 0.285 = 0.715

Rounded to four decimal places, the probability is 0.7150.

3. The probability that Professor Jane is funded but Professor Joe is not is the product of the probability of Jane getting funded and the probability of Joe not getting funded.

P(Jane gets funded and Joe does not) = P(Jane gets funded) * (1 - P(Joe gets funded)) = 0.62 * (1 - 0.25) = 0.62 * 0.75 = 0.465

Rounded to four decimal places, the probability is 0.4650.

4. Given that at least one of the professors is funded, the probability that Professor Jane is funded but Professor Joe is not can be found using the conditional probability formula:

P(Jane funded | at least one funded) = P(Jane funded and Joe not funded) / P(at least one funded) We have already found these probabilities in steps 2 and 3. So we can plug those in:

P(Jane funded | at least one funded) = 0.465 / 0.715 = 0.6503

Rounded to four decimal places, the probability is 0.6503.

Extra:

When we talk about probabilities, we’re referring to the likelihood that a certain event will happen. Probabilities range from 0 (the event definitely will not happen) to 1 (the event definitely will happen). When events are independent, like in this question, the outcome of one event does not affect the outcome of the other. This is why we can simply multiply the probabilities of the two events to find the combined probability.

The concept of "at least one" is commonly mistaken for needing to add probabilities, but in this case, it's easier to calculate the opposite (neither happening) and subtract from 1, because "at least one" includes all scenarios except the one where neither event happens.

Finally, conditional probability is the probability of an event happening given that another event has already happened. It is calculated by dividing the probability of both events happening by the probability of the event that has already happened.