Psychological Probability Quiz

Marcus and Catti's Kids

Everyone knows it's lucky to be born in Esmerel. Marcus and Catti just had their sixth child, and the mathemagician sees the opportunity for another probability quiz. Since you don't know Marcus and Catti's family, he asks you which of these birth orders you think is more likely:
  1. Girl Boy Girl Boy Boy Girl
  2. Boy Girl Boy Boy Boy Boy
  3. Either is equally likely

Answer

Assuming that boys and girls are equally likely, the answer is 3: Either is equally likely.

Actually, slightly more boys are born than girls, so the completely correct answer is that the order with more boys is slightly more likely. If you know Marcus and Catti, you will know that in this case, number 2 is correct, and that they just had their fifth lucky boy.

But most people, says the mathemagician, pick answer number 1, and for an interesting reason. People base their answers on heuristics, rules of thumb. (Heuristics are "rules" that aren't guaranteed to work, but much of our thinking is based on them. They become our heuristics because they have worked, or we believe they have worked, for us in the past.)

In this case, the heuristic which gives the wrong answer is known as the Representativeness Heuristic. In other words, we think that things that are random should LOOK random.

A better-known heuristic is the Vividness Heuristic, in which vivid events are judged more probable than they really are. So most people perceive that there is more chance of dying in an airplane crash than in a car accident, even though dying in a car accident is far, far more likely.

The Bank Teller

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more likely:
  1. Linda is a bank teller?
  2. Linda is a bank teller who is active in the feminist movement?

Answer

The answer is 1: Linda is a bank teller.

Probability Rule #3 (see the Normative Rules) tells us that:

The probability of answer 2 can never be greater than the probability of answer 1. This wrong answer is so frequent, though, that maybe people perceive 1 as implicitly saying that Linda is NOT in the feminist movement. Or maybe we average the two probabilities instead of adding them.

Even when people are given unrelated possibilties, such as:

  1. What is the probability that it will rain tomorrow?
  2. What is the probability that municipal taxes will go up this year?
  3. What is the probability that it will rain tomorrow and that municipal taxes will go up this year?
If the second choice is seen as likely, most people rate number 3 as the most likely of all!

The Hospitals

A certain town is served by two hospitals. In the larger hospital, about 45 babies are born each day. In the smaller one, about 15 babies are born each day. Although the overall proportion of girls is about 50%, the actual proportion at either hospital may be greater or less on any day. At the end of a year, which hospital will have the greater number of days on which more than 60% of the babies born were girls?
  1. the large hospital
  2. the smaller hospital
  3. neither -- the number of these days will be about the same

Answer

The answer is 2: the smaller hospital.

The Law of Large Numbers (see the Normative Rules) says:

In this case, the number of children born each day is a random sample of the number of children born in a year. There is much more deviation in a small (daily, in this case) sample than over an entire year. Similarly, there will be more deviation in a sample of size 15 than in one of size 45. Even though both hospitals will have about 50% girls over the year, the smaller hospital will have more days in which the number is higher than 60% (and more days when it is less than 40%). (Consider the extreme case of a hospital that has one baby a day. Every day will be either more than 60% girls (one) or less than 40% girls (zero).

(Exercise for the reader: Actually the proportion of girls is about 49%. How does this affect the answer?)

People commonly and mistakenly believe that even very small samples will have the same composition as large ones. Remember this next time someone uses a small survey sample to "prove" something -- the results may be invalid!

No one born in this century has lived to be a hundred.
- Numbers for the real world: The numeracy page.
- Tricks with numbers.
- Where did the dollar go?
- Wordlore and wordplay.
- Return to the front gate of Esmerel.
Page maintained by the Mathemagician.