Say that a coin is flipped 10 times. What is the probability that the first 3 flips are all heads provided that there are an even number of heads and tails all together?
- I understand that if we simply flip 3 coins, the probability that 3 heads appear is 1/6. How do we take into account the fact that 5 of 10 are definitely heads??
Flipping a coin three times gives 8 possible outcomes only one of which is “HHH”.
Therefore the probability of “HHH” is 1/8 not 1/6.
Thus the same idea gives an approach to the question.
If in ten flips we have an even number of heads, 0 2 4 6 8 10, then we have an even number of tails.
In ten flips that can happen in ways.
Now we must find out how many of those begin the string with “HHH…”.
If we already have three H’s then in the next seven places we must have 1, 3, 5, or 7 H’s to have an even number of heads and tails all together.
The number of ways that can happens is .
Simply divide the two.