What is the probability of having a girl four times in a row?

One way to solve this problem is to set up the sample space as the set of all possible sequences of births. For example, one possible sequence is (G,B,G,B), where you get Girl followed by Boy followed by Girl followed by Boy. Overall, there are sixteen possible sequences:

(G,G,G,G), (G,G,G,B), (B,B,B,G), (B,B,G,G), (G,B,G,G), (G,B,G,B), (G,B,B,G), (G,B,B,B), (B,G,G,G), (B,G,G,B), (B,G,B,G), (B,G,B,B), (B,B,G,G), (B,B,G,B), (B,B,B,G), (B,B,B,B)

Of these sixteen sequences, only the first sequence has four Girls. Since each sequence would seem to have an equal probability, the logical inference is that the probability of getting four straight Girls is 1/16.

What is the probability of getting exactly three Girls in four births? Now, the calculation is not so simple. It turns out that of the sixteen sequences, four of them have three Girls. The general answer to this question was given by Pascal.

Pascal's Triangle

1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

Each row of the triangle is constructed by adding the adjacent numbers in the preceding row, and then putting a one on the far left and far right.

For example, the second row is 1 2 1. When we add 1 and 2, we get 3, which we put in between them on the third row. Outside the two 3's, we extend the row by putting a 1 to the left and to the right.

Question: What will be the contents of the sixth row of the triangle ?

How else might we compute the probability of getting four Girls in a row? One approach is to think in terms of a sequence of births, where as soon as you get a Boy, you stop.

1. The probability of being a girl on the first birth is 1/2. If you have a Boy on the first birth, you might as well stop, because you cannot possibly get four Girls. So, half the time you stop, and half the time you keep going.
2. Assuming we kept going, then we have another baby. Again, the probability of having a girl is 1/2. Again, we only keep going if we have a Girl. So half the time we keep going. Overall, the chance that we will keep going is 1/2 of 1/2, or 1/4.
3. By now, 3/4 of the time we will have stopped, and 1/4 of the time we will have moved on to the next birth. Again, the probability of a Girl is 1/2. So, the probability that we will keep going is 1/2 of 1/4, or 1/8.
4. Finally, we have the fourth birth. We only get to this point 1/8 times. Again, the probability of a girl is 1/2. The probability of four Girls is thus 1/2 of 1/8, or 1/16.

As a shortcut, we could say that the probability of having Girls on any one throw is 1/2. The probability of getting four Girls in a row therefore is (1/2)(1/2)(1/2(1/2), or (1/2)⁴.

A general approach to analyzing baby gender is called Pascal's triangle. The triangle is a shortcut way to describe the sample space for the number of Girls and Boys from a sequence of births. The first row says that with one birth, we can have either all Girls (1) or all Boys (1).

The second row says that if we have two babies, we have one chance of getting all Girls, two chances of getting one Girl and one Boy, and one chance of getting all Boys.

The third row says that if we have three babies, we have one chance of getting all Girls, three chances of getting one Girl and two Boys, three chances of getting two Girls and one Boy, and one chance of getting three Boys. Since the sum of the row is 8, the probability of getting two Girls and one Boy is 3/8.

Question: use the triangle to find the probability of getting exactly three Girls in four births. What about exactly three Girls in six births?

Implicitly, we are relying on the assumption that each birth is independent. When births are independent, it means that the probability of a baby being a girl does not depend in any way on previous births.

(Same probability works for determining having a boy four times in a row)

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