A state senator believes that 25 of all senators on the...

A state senator believes that 25% of all senators on the Financial Committee will strongly support the tax proposal she wishes to advance. Suppose that this belief is correct and that five senators are approached at random. a) What is the probability that at least one of the five will strongly support the proposal? _________ (1) b) What is the probability that a majority of the five will strongly support the proposal? _________ (1) c) Find the expected value of number of senators that will strongly support the proposal. _______ (1) d) Find the standard deviation of number of senators that will strongly support the proposal.

Transcript

00:01 Here we are looking at a random variable, s, which is defined as the number of senators who strongly support our proposal when we pick five of them at random.

00:15 And the condition here is that the probability that any randomly selected senator supports our proposal is 1 over 4.

00:24 We want to ask some things about this random variable s.

00:28 And the first one is what is the probability that s is at least one? so the way i would want to solve this is say this is actually 1 minus the probability that s is less than 1.

00:45 And why would i want to do that? because that's pretty easy to compute.

00:51 That's equal to 1 minus.

00:53 I mean, if s is less than 1, it has to be 0.

00:57 So what's the probability that s is zero? well, that's just 3 over 4 to the 5th.

01:05 The probability that a senator does not strongly support our proposal is 3 over 4, and then the probability that all 5 that we've chosen don't is 3 over 4 to the 5th.

01:21 This is going to be equal to 1 minus 243 over 1024.

01:28 It's a bit messy, but we can revert it as 781 for 1024.

01:35 And then of course you want something that looks like a probability.

01:38 This is roughly 0 .76.

01:42 So the probability that at least one senator will support our proposal is actually pretty good.

01:50 Of course, we are approaching five.

01:54 The next thing we want to determine is what's the probability that at least three senators support.

02:00 So here i'm going to go more straightforward.

02:04 You know, we're just going to check what's the probability that three, four, and five senators support the proposal.

02:13 So first, exactly three senators support the proposal.

02:17 If we pick some three of them, and then we're going to take, what's the probability that three of them do support? that's one over four cube.

02:26 And then the other four don't support, so that's three over four squared.

02:31 Okay, what about the probability that exactly four senators? so five and three is up to the down.

02:38 I'm sorry, i got that.

02:39 Five choose three.

02:41 The next case is five choose four...

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