A high school student is anxiously waiting to receive mail...

A high school student is anxiously waiting to receive mail telling her whether she has been accepted to a certain college. She estimates that the conditional probabilities; given that she is accepted and that she is rejected, of receiving notification on each day of next week are as follows: She estimates that her probability of being accepted is $.6 .$ (a) What is the probability that mail is received on Monday? (b) What is the conditional probability that mail is received on Tuesday given that it is not received on Monday? (c) If there is no mail through Wednesday, what is the conditional probability that she will be accepted? (d) What is the conditional probability that she will be accepted if mail comes on Thursday? (e) What is the conditional probability that she will be accepted if no mail arrives that week?

A high school student is anxiously waiting to receive mail telling her whether she has been accepted to a certain college. She estimates that the conditional probabilities; given that she is accepted and that she is rejected, of receiving notification on each day of next week are as follows:
She estimates that her probability of being accepted is $.6 .$
(a) What is the probability that mail is received on Monday?
(b) What is the conditional probability that mail is received on Tuesday given that it is not received on Monday?
(c) If there is no mail through Wednesday, what is the conditional probability that she will be accepted?
(d) What is the conditional probability that she will be accepted if mail comes on Thursday?
(e) What is the conditional probability that she will be accepted if no mail arrives that week?

Key Concepts

Conditional Probability

This concept refers to the probability of an event occurring given that another event has occurred. It is fundamental in situations where the outcome of one event affects the probability of another and is used to update our knowledge about an uncertain situation based on new information.

Law of Total Probability

The law of total probability allows one to compute the overall probability of an event by considering all the possible ways that event can occur, each weighted by the probability of the scenario in which it occurs. This is essential in situations where an event can occur under several different conditions or partitions of the sample space.

Bayes' Theorem

Bayes' Theorem provides a mechanism for updating probabilities as new data becomes available. It relates the conditional and marginal probabilities of random events and is particularly useful for revising predictions or hypotheses in light of observed evidence.

Sequential Probability Updating

This concept involves revising probability estimates as additional evidence or lack of evidence accumulates over time. By incorporating new information step by step, one can refine the probability of a particular outcome, which is critical in dynamic scenarios where outcomes unfold over time.

Transcript

00:01 Here we have a high school student who is anxiously waiting to receive mail telling her whether she's been accepted to a certain college.

00:06 And she estimates the conditional probabilities of receiving notification on each day of the week, given that she is accepted and given that she's rejected and that's in the table.

00:15 So she estimates that her probability being accepted is 0 .6.

00:19 And for part a, we want to know what's the probability that she receives mail on monday.

00:23 And so we're going to let a and r denote acceptance and rejection, and then m.

00:31 Monday, t.

00:32 Tuesday, w.

00:33 Wednesday, t .h.

00:33 Thursday, and f.

00:34 Friday.

00:36 So we want the probability that she receives mail on monday.

00:39 And since a and r mutually exclusive and exhaustive events, this is going to be the probability of mail on monday is equal to probability of mail and monday in accepted, terms of the probability of acceptance plus probability of male and rejections.

00:57 On monday times the probability of rejection.

01:02 And so we can just plug in the values from the table because we have those in the table, whether mail slash accepted and mail slash rejected.

01:11 So this will come out to 0 .11.

01:16 And then for b, we're asked, what's the conditional probability that she receives mail on tuesday given that she does not receive mail on monday? and so this is going to be the probability of mail on tuesday, given that she does not receive mail...

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