multistage experiments

Question

#1. In spite of experimental evidence to the contrary, one of your students still maintains that the probability of getting one head and one tail when two coins are dropped is 1/3. Explain how you think the student has arrived at this conclusion and what you would do to help him reach the correct theoretical probability.

Answer 1

The 50-50 probably for coins might lead one to the conclusion that it's like a jar containing a mix of "head coins" and "tail coins". Suppose there is just one of each: On the first draw, you have a probability of 1/2 that it will be heads. The other draw is guaranteed to be tails, and the net odds of drawing one head and one tail is 1.

With 4 coins (2 heads, 2 tails), the probability of 1 heads/1 tails is 1/2 * 2/3 = 1/3

(regardless of what is drawn first, there are 2 left of the opposite--out of 3)

The student's fallacy is simply that the events are connected. The best way to demonstrate it might be to flip 2 coins at the same time and prove that 1/2 of the tries will result in a head-tail pair.