4. (15 pts) Consider a coin. Assume that, when we toss it...

4. (15 pts) Consider a coin. Assume that, when we toss it once, the probabilities on the outcomes H, T are given by $P({H}) = p$ and $P({T}) = q$, where $p, q > 0$ and $p + q = 1$. Now, we toss the coin three times and asume that one or two tossings does not affect the rest. Consider the following three events. • A: The first coin lands heads. • B: The second coin lands tails. • C: Coin lands heads exactly once among three tossings. (a) (3 pts) Show that the events A, B are not mutually exclusive, but independent. (b) (9 pts) Show that $pq ? 1/4$ and use this fact to show that the events B, C are not independent and also the events A, C are not independent. * These imply that A, B, C can be independent. (c) (3 pts) Show that $P(A?B?C) ? P(A)P(B)P(C)$.

Transcript

00:02 So in this question we have that a and b are independent, and that means the probability of a times the probability of b is the probability of a and b both happening.

00:13 We also have that the probability of a is less than or equal to the property of b.

00:18 Now for part one, we have that the probability of a and b is 1 in 15.

00:26 And what that tells us is that the probability of a times the probability of b is 1 over 15.

00:34 And we have the probability of neither of them happening, so the probability of not a and not b, is 8 in 15.

00:44 So that tells us that the property of...

00:48 So the property of not a and not b is 1 minus the property of a times 1 minus the property of b.

00:57 And that's 8 in 15.

00:59 So we get 1 minus the property of a plus the property of b, plus the property of a times the property of b.

01:08 And that's 18 and 15.

01:14 So that tells us that the probability of a plus the probability of b is equal to 1 plus 1 over 15 minus 8 over 15.

01:25 So that's going to be 1 minus 7 over 15 so as 1 minus 6 over 15 even, which is 9 over 15.

01:43 Sorry, yeah, this is 8 over 15 anyway.

01:47 1 plus 1 over 15 minus 8 over 15 is 8 over 15.

01:52 So we've got this equation up here, and we've got this equation down here.

01:57 So what we can do is we can just try things.

02:00 So let's try factors of 15.

02:03 So we can try something like the probability of a is 1 5th, and the probability of b is 1 3rd, because we know when we multiply these together, we get 1 in 15.

02:14 But what happens when we add them together? 1 5th plus 1 3rd, well, wouldn't you know it, that's 8 over 15? so this satisfies both equations, satisfies both equations.

02:30 So that means we're done.

02:34 That gives us our probability of a and our probability of b.

02:37 Okay, now the second part, the probability of exactly one of them happening is 13 over 56.

02:45 So that's the probability of a happening and b not happening, plus the probability of a not happening and b happening.

02:53 This is 13 over 56.

02:56 So let's do a little bit of work on this.

02:58 So this is the probability of a times one minus the probability of b, plus one minus the probability of a times the probability of b, and that's 13 over 56.

03:10 So what do we get here? well, we get the probability of a plus the probability of b.

03:16 We get minus the probability of a times the probability of b times 2, and this is 13 over 56.

03:30 And the probability of neither of them happening is three quarters.

03:35 So remember that the probability that neither of them happening, we worked out was one minus some of the properties plus the product.

03:48 Sorry, this should be plus...

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