A bag has 20 marbles 5 white 4 lime and 2 yellow 9 teal we...

A bag has 20 marbles (5 White, 4 Lime, and 2 Yellow, 9 Teal). We remove 4 of these uniformly at random (all options being equally likely, without replacement). (a) What’s the chance they are all different colors? (b) What’s the chance they are all different colors given that at least one of them is Yellow? (c) What is the expected number of Teal marbles selected?

Transcript

00:01 We assume that in the back there are x number of green marbles.

00:04 So the green marbles, it is x.

00:06 And the blue marbles, it is white.

00:09 Totally stale, so we can write x are red to y.

00:13 This will be 10.

00:14 It is given that blue marbles are greater than green ones.

00:17 So we can write y greater than x.

00:19 Now for the first part, we are taking two marbles at a time and the probability of getting both of them of same color, it will be 24 divided by 45.

00:29 So both of them to be same color, either we take two green or two blue if we are taking two green both of them are green the probability will be xc2 divided by 10c2 similarly if we take two blue the probability will be yc2 divided by 10c2 so the addition will give us the value 24 divided by 45 so the solution will give xc2 divided by 10c2 added to yc 2 divided by 10 c2 that will be 24 divided by 45 we can simplify and multiply with 10 c2 and the value of x c2 and y c2 this will give us x multiplied with x minus 1 divided by 2 added to y multiplied with y minus 1 divided by 2 will be equal to 24 divided by 45 multiplied with 10 c2 value of 10 c2 that will be 10 multiplied with 9 divided by 2 which will give us 45 further simplifying we get x squared x added to y square minus y that will be 24 multiplied to which is 48 now we can further simplify x squared to y square minus minus of x plus y which will give us minus 10 as the value of x plus y it is 10 so we are using actually x plus y that is equal to 10 so x square plus y square this becomes 58 now we take these two this one and x plus y equal to 10 we can simplify this two this would give us the value x being equal to 3 y being equal to 7 in this particular case we took y is equal to 7 not 3 because we had the value y greater than x this was the condition given that there are more blue marbles than green ones therefore the number of green marbles it is 3 number of blue marbles it is 7 in the second in the second in the second part, we are taking three marbles at a time.

02:43 In this case, we are trying to find most likely number of blue marbles.

02:47 If you are trying to find blue marbles, we can take one blue marble, two blue marble, or all of them three blue marbles...

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