On independence · Mar 12, 12:55 PM

We’re having a quiz in CP Stats, tomorrow. On the quiz, I’ll give you one scenario. You will need to verify that the trials are Bernoulli trials (that is, like coin flips), prepare a probability table, and use your table to state probabilities. You’ll also calculate the expected value (or mean), the variance and the standard deviation of the binomial random variable.

The question most likely to trip students up is the question of independence. Like a coin flip, the outcome of a Bernoulli trial must not depend on what’s happened prior to it. Loosely speaking, we’ve said that a coin has no memory, and it certainly does not. The outcome of the next coin flip does not depend on the 50 or 100 coin flip outcomes that came before.

We must be careful not to overstretch this analogy, though. Literally speaking, a card in a deck has no memory, either. However, outcomes may depend on what’s come before. Consider the following probability question:

If I draw 20 cards out of a deck and I do not replace them, what’s the probability that the 21st card drawn is a red card?

Hopefully, you wouldn’t claim to know the answer to this one because so many cards have been drawn out of the deck. Your answer depends on the cards previously withdrawn. If all 20 cards drawn previously were red, the probability that the 21st is red is low (6/32 or 3/16). Draw out 20 black cards and the probability of a red on the 21st draw is much higher — 13/16.

So, be prepared to determine whether too many cards (or other things, like people) are drawn out of the population. For this purpose, we’ll say that “too many” is more than 10% of the population. In the context of our card drawing problem, we cannot draw out more than 5 cards (5% of 52 is 5.2) and still treat each outcome as being independent.