How do you do part d of this problem: Bob spends the majority of a certain eveni

Question

How do you do part d of this problem:

Bob spends the majority of a certain evening in his favorite drinking establishment. Eventually, he decides to spend the rest of the night at the house of one of his two friends, each of whom lives ten blocks away in opposite directions. However, being a bit intoxicated, he engages in a so-called “random walk” of n = 10 blocks where, at the start of each block, he first either turns and faces due west with probability 0.4, or independently, turns and faces due east with probability 0.6, before continuing. Using this information, answer the following. Hint: Let the discrete random variable X = “number of east turns in n = 10 blocks.” (0, 1, 2, 3, …, 10) (a) Calculate the probability that he ends up at Al’s house. (b) Calculate the probability that he ends up at Carl’s house. (c) Calculate the probability that he ends up back where he started. (d) How far, and in which direction, from where he started is he expected to end up, on average? (Hint: Combine the expected number of east and west turns.) With what probability does this occur?

Explanation / Answer

This is an example of binomial probabilitydistribution.

We find the expected value for each of the possible combination of steps that he might have taken.

Then, we calculate the probability of these combinations and multiply it with the net movement of steps in the east direction.

The sum of the expected values gives the average movement of the person in the east direction.

The calculations are shown below.

Thus, on an average, Bob will land up to blocks on the east.

Hope this helps.

0.6 0.4 Combination East West Probability Net-East Expected value 1 0 10 0.000104858 -10 -0.0010486 10 1 9 0.001572864 -8 -0.0125829 45 2 8 0.010616832 -6 -0.0637010 120 3 7 0.042467328 -4 -0.1698693 210 4 6 0.111476736 -2 -0.2229535 252 5 5 0.200658125 0 0.0000000 210 6 4 0.250822656 2 0.5016453 120 7 3 0.214990848 4 0.8599634 45 8 2 0.120932352 6 0.7255941 10 9 1 0.040310784 8 0.3224863 1 10 0 0.006046618 10 0.0604662 Sum 2

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