9. Linda took a multiple choice test. Each questionhad 4 possible answers from which to choose.There were 6 questions where she had to guess. Make up a probability distribution for this exeriment.

• a) Make up a probability disrtrbution for this experiment.
• b) Draw the graph.
• c) What is the probability that she got them all right?
• d) What is the probabity that she got them all wrong?
• e) What is the probability that she got at least 4 right?
• f) What is the probability that she got at least 4 wrong?
• g) What is the expected number of correct guesses?

The experiment is guessing on an answer, and, in this case, it is being repeated 6 times. If ech quiestion has 4 possible answers, the probability of getting an answer correct by randomly guessing is 1/4 or 25% on each question. The probaility of getting x answers righ by guessing is

P(x) = C(n, x)pxqn-x

where   q = 1 - p.   The probability distribution, then, looks like

b) So the graph looks like

c) The probability of getting them all right is   1/4096,   which from the graph, we see is what they call a vanishingly small probability.

d) The probability that she got them all wrong is   729/4096.

e) To find the probability that she got at least 4 right, add up the probabilities of all the outcomes in the event. The event that she got at least 4 right is

{4, 5, 6}

Look up their probabilities in the distribution.

and add them up. The probability is   154/4096 = 77/2048.

f) The event of getting at least 4 wrong is

{0, 1, 2, 3}

Notice that getting at least 4 wrong is the complementary event from getting from getting at least 4 right, so the probabilities of these two events add up to one.

g) To get the expected number of correct guesses, multiply the probability of getting that many correct by the number of correct.

6144/4096 = 2048x3/2048x2 = 3/2 = 1.5

Or one could use the formula that the expected number of successes is

np = 6x(1/4) = 3/2.

Notice that the graph peaks out above   x = 1.5.