Write your answer to at least 3 decimal places if...

Write your answer to at least 3 decimal places if appropriate. 1. Suppose a jar contains ten balls: six red and four white. Two balls are then randomly selected without replacement from the jar, and the number of red balls is recorded as the outcome of a random variable X. Which of the following statements is correct? X doesn't have a binomial distribution since there is an unequal number of red and white balls in the jar. For this reason, the probability p of choosing a red ball changes from trial to trail. X ~ B(10, 0.6) since the balls are selected without replacement. X ~ B(6, 0.4) since there is an unequal number of red and white balls in the jar. X doesn't have a binomial distribution since the balls are selected without replacement. For this reason, the probability p of choosing a red ball changes from trial to trail. 2. In a bag, there are 9 pieces of chocolate in total: 5 pieces of them are milk chocolate and 4 pieces are dark chocolate. Suppose that you have no preference for either and simply pick two pieces of chocolate out of the bag completely at random. Let X be the number of milk chocolate from your draw. (a) Compute P(X = 1) (b) Compute the mean of X. (c) Compute the variance of X. 3. Sam is a very good darts player and can hit the bull's eye (red circle in the centre of the dart board) 60% of the time. Calculate the following probabilities: (a) What is the probability that he hits the bullseye for the 10th time on the 15th try? (b) What is the probability that he hits the bullseye 10 times in 15 tries? (c) What is the probability that he hits the first bullseye on the third try? 4. Houying and Thomas are going to play 10 games of Mario Kart against each other. Both Houying and Thomas are experienced players so their chances of winning don't change from game to game. Houying is a better player however and has a 60% chance of winning any game against Thomas. You can assume all the games are independent. In the initial 6 games, Houying won 5 of them. Compute the probability that Houying will win 7 games in total out of 10, given Houying has already won 5 out of the initial 6 games. Compute the required probability.

Transcript

00:03 So we have a question that says the jar contains 10 balls, and then in them we have six red balls, and then we have four white balls.

00:15 And then two balls are drawn at random without replacement from the jar, and the number of red balls is recorded as the outcome of random variable x.

00:22 So x is the number of red balls of r out of the two drawn.

00:37 So this is not going to be a binomial distribution because this is the number of red balls of r out of the two drawn.

00:42 Probabilities change.

00:43 Because if we're looking for the district, because questions about the distribution of x, and this is not binomial.

00:54 And the reason for that is because when you draw one of the balls out, let's say you pick one of the two balls out and you pick a red ball, there are no longer 10 in there.

01:02 There are nine.

01:03 And then there is no longer six red balls.

01:05 There are now five.

01:06 So your probabilities for the next draw would be different.

01:10 And a binomial distribution requires that the probability be the same.

01:13 So it's not binomial.

01:14 No meal.

01:15 All right, so number two is a similar scenario, except we have nine pieces of chocolate.

01:28 And there's nine.

01:32 And then we're told five of them are milk chocolate.

01:40 And then we're told four of them are dark chocolate.

01:44 And we have no preference on either and simply pick two pieces out of the bag.

01:48 And x is the number of milk chocolate.

01:50 We have x is the number of milk chocolates.

02:01 And we want to compute the probability that x is one, that we get one out of the two balls.

02:11 Sorry, thinking about the first question.

02:14 One out of the two pieces of chocolate is milk chocolate.

02:19 So let's think about what we have.

02:21 We have the first draw, if you want one of the two, if you have two, you have two, two choices.

02:31 You could have this first one, let's see you pick a milk chocolate here.

02:34 Well, that's going to be five out of nine.

02:38 But then the second one, if that's the milk chocolate, then this one has to be dark chocolate.

02:42 So that means it would be four out of eight because there are now eight pieces in there.

02:46 You could also flip it around.

02:48 You could have also picked four out of nine.

02:51 The first one could have been dark chocolate.

02:52 And then the second one would be milk chocolate, but that'd be five out of eight.

02:57 And so you're going to add these two probabilities together, and that's going to be the probability that x is one.

03:00 And they have that right up here.

03:05 Here's a distribution which is going to deal with the next question anyway.

03:09 One is right here.

03:12 It's four nights times five -eighths times two.

03:14 And that's the same thing as adding these two together.

03:16 And we get 0 .5 repeating, or it ends up being 5 -9ths.

03:25 All right.

03:26 Now we want to compute the mean and variance of x.

03:29 So the mean is calculated as the sum of all the x values multiplied by their respective probabilities.

03:40 And i'm going to use the symbol.

03:43 I mean, i'm going to use the expected value.

03:45 Let's talk about an expectation here.

03:48 So the expected value is the mean, and that equals the sum of x times p of x values.

03:51 And then while we're here, we'll talk about the variance because we're going to do that in a second.

03:54 The variance of x is calculated as the expected value of x squared minus the expected value of x quantity squared.

04:04 So we're taking the mean and squaring it here.

04:05 We're going to find that out.

04:06 So what's this part, the expected value of x squared, well, that's going to be equal to the sum of all the x squared values, or x values squared, multiplied by their probabilities.

04:19 So let's go ahead and get this underway here.

04:26 So i've got those right here.

04:28 We had to go ahead and figure, i'll show you the whole thing, we had to go ahead and figure out the rest of these probabilities.

04:33 So for there to be zero chocolates, zero milk chocolates, it had to be four ninths, that's a dark chocolate times the next dark chocolate, which would be three eighths.

04:43 So it ends up being a sixth.

04:47 And then for two, that means you have five -nights, that's a milk chocolate.

04:52 And then the other one, four -eighths, that's the next milk chocolate.

04:55 After you pick one of the milk chocolates here, there are four -left, and then you've taken one at the total, which means four -eights.

05:01 So that's 0 .277.

05:05 Now, to find the mean, x times p of x, x times p of x, take all those values, summed them together...

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