Expected value

(a) The expected value of the number shown when we draw a single slip of paper is given by the weighted average of the possible outcomes, where the weights are the probabilities of each outcome. Let X be the random variable representing the number shown on the slip of paper. Then we have:

X = { 3, with probability 8/10 = 0.8 5, with probability 2/10 = 0.2 }

So, the expected value of X is:

E[X] = 3(0.8) + 5(0.2) = 3.4

Therefore, the expected value of the number shown when we draw a single slip of paper is 3.4.

 

(b) If we add one additional 3 to the bag, then the number of slips with a 3 on them increases to 9, while the number of slips with a 5 on them remains at 2. The probabilities of each outcome are now:

X = { 3, with probability 9/11 5, with probability 2/11 }

So, the expected value of X is:

E[X] = 3(9/11) + 5(2/11) = 3.182

Therefore, the expected value of the number shown when we draw a single slip of paper after adding one additional 3 to the bag is 3.182.

 

(c) If we add two additional 3's to the bag, then the number of slips with a 3 on them increases to 10, while the number of slips with a 5 on them remains at 2. The probabilities of each outcome are now:

X = { 3, with probability 10/12 = 5/6 5, with probability 2/12 = 1/6 }

So, the expected value of X is:

E[X] = 3(5/6) + 5(1/6) = 2.833

Therefore, the expected value of the number shown when we draw a single slip of paper after adding two additional 3's to the bag is 2.833.