An american roulette wheel contains 38 numbers 18 are red 18...

An American roulette wheel contains 38 numbers: 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally likely to land on any of the 38 numbers. Suppose that you bet $1 on red. If the ball lands on a red number, you win $1; otherwise you lose your $1. Let X be the amount you win on your $1 bet. Find the expected value of the random variable X . Interpret the expected value. Approximately how much would you expect to lose if you bet $1 on red 100 times? 1,000 times? Is this game fair for the players? Explain.

Transcript

00:01 So we're talking about roulette.

00:02 Roulette has 38 different possibilities.

00:06 18 are red, 18 are black, and two are green.

00:16 Now, your payouts, if you, and you're obviously, they're equally likely to land on any of these 38 different potential landing spots.

00:27 The payouts are that you earn $1 if you get it right, or you earn one if you get it right, and then you lose $1 or lose your bet if you get it wrong.

00:39 So it's an equal payout as it is compared to you'd lose.

00:44 So the question is, what's our expected value? how much do we expect to lose if we bet a dollar on red? and then let's ramp it up, you know, 100 times, 1 ,000 times, etc.

00:54 So our expected value is going to be, all right, the expected, let's see if i can write this correctly, expected value of x, which is the amount you win on your bet, is equal to the probability of winning plus like the probability of losing, like those two together.

01:23 So our probability of winning, we have 18, if we are always looking at right, we have 18 out of 38.

01:31 And we know that we're, we'd be earning a dollar.

01:34 So we'd multiply that by $1.

01:38 The probability of losing is a little bit higher.

01:40 It's the other 18 black and the two greens.

01:43 So that's 20 out of 38.

01:46 And then we're multiplying that by negative $1 because it means you're losing a dollar.

01:52 So 18 out of 38 times $1.

01:57 We can simplify this.

01:58 First just to keep it in fractions.

02:01 We end up with 9 out of 19.

02:06 And then when we're adding this 20 out of 38, we're multiplied by negative 1.