December 2003

Vol. 5, No. 12


Quality Quiz
from Professor Cleary

Congratulations! 
You're right!

While Simsack contemplates the imprint of his shoe on the carpet, Dr. Stan Deviation reminds the class of the multiplication rule he had presented last week using a flip of a coin, asking them what the probability is of getting a head on the first flip and a head on the second flip. The table below indicates the complete set of outcomes for two flips of a coin.

One way to see this is to observe that there are four possible outcomes (E1, E2 E3,E4). E1 is one of the four: Ň, or 25 percent. The second way to discover this is to apply the multiplication rule: P(A and B) = P(A) x P(B)

or

P (H and H) = P(H) x P(H)

= .5 x .5

= .25

Looking back at the theoretical histogram, imagine cutting these distributions in half, and think of the left side and right side. The probability of randomly getting an observation on either the left side or the right side would be 50%, in the same way that flipping a coin and getting a head or tail is 50%. Therefore, for a sample size of 2, the probability of getting two observations from the left side would be (.5)(.5) = .25, or 25 %.

P (L and L) = P(L) x P(L)

P (L and L) = .50 x .50

P (L and L) = .25

He tells the class to look again at the probability tree and then asks about the probability of getting one head and one tail.

They examine the tree and note that this happens when the first flip is a head and the second is a tail, or E2. It also happens when the first flip is a tail and the second flip a head, or E3. Thus in two events out of four, this will happen 2/4, or 50 percent of the time. He reminds them that they have applied the addition rule, or

    P(A or B) = P (A) + P(B)

P(HT or TH) = P(HT) + P(TH)

P(HT or TH) = Ň + Ň

P(HT or TH) = ż or .50

As in the example above, this thinking can be converted from flipping a coin to getting an observation from the left side or the right side of the theoretical distribution, thus:

P(LR or RL) = P (LR) + P(RL)

P(LR or RL) = .25 + .25

P(LR or RL) = .50

With all this, the trimodal distribution can be explained when sample size is two.

Viewing the trimodal distribution, it is clear that the large center of the distribution is made of LR or RL, which occurs 50 percent of the time. The smaller hump to its left is made up of LL, which occurs 25 percent of the time, and the smaller hump to the right is made up of RR, which occurs 25 percent of the time.

This all might appear to be merely academic drivel to you. But hang on—next month's quiz will demonstrate how this understanding offers a partial explanation for why control charts work. Now that's important.

Simsack was taking notes for this session—a good thing, since he had missed the mark entirely.

Click here to view a short video related to this quiz.


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