Thread: Coin tossing problem I have trouble understanding

So the question is:

100 coins are tossed, what is the probability that exactly 50 are heads?

Now the answer is... there are 2^100 outcomes
and 100_C_50 of those outcomes have exactly 50 heads (using generating functions for example)

Hence the probability is 100_C_50/2^100 which is approximately 0.08


But what I don't understand is that why does order matter in this question? Why isn't the answer 1/101. You see there are 101 distinct sets' of outcomes. 1 set is all heads, 1 set is all heads except one, ..., 1 set is all tails. Among these sets, there is one where 50 are heads and 50 are tails.

Hence that is 1 set out of 101 sets. So the answer would be 1/101.

What is wrong with this reasoning?
If instead of 100 coinS, it was one coin tossed 100 times--would the answer still be 100_C_50/2^100?

Please help me, I was never taught probability in school and trying to learn it on my own!

Originally Posted by vahshi

100 coins are tossed, what is the probability that exactly 50 are heads?
If instead of 100 coinS, it was one coin tossed 100 times--would the answer still be 100_C_50/2^100?

Conceptually there is no difference in tossing one coin 100 times and tossing 100 coins at one time. You are simply counting the number of heads. Therefore, order has nothing to do with it.

Originally Posted by vahshi

Why isn't the answer 1/101. You see there are 101 distinct sets' of outcomes. 1 set is all heads, 1 set is all heads except one, ..., 1 set is all tails. Among these sets, there is one where 50 are heads and 50 are tails.

Where in the world does 101 come from?
There are 100 places in which there either H or T.
That is possible outcomes.

Originally Posted by vahshi

So the question is:

100 coins are tossed, what is the probability that exactly 50 are heads?

Now the answer is... there are 2^100 outcomes
and 100_C_50 of those outcomes have exactly 50 heads (using generating functions for example)

Hence the probability is 100_C_50/2^100 which is approximately 0.08


But what I don't understand is that why does order matter in this question? Why isn't the answer 1/101. You see there are 101 distinct sets' of outcomes. 1 set is all heads, 1 set is all heads except one, ..., 1 set is all tails. Among these sets, there is one where 50 are heads and 50 are tails.

Hence that is 1 set out of 101 sets. So the answer would be 1/101.

What is wrong with this reasoning?
If instead of 100 coinS, it was one coin tossed 100 times--would the answer still be 100_C_50/2^100?

Please help me, I was never taught probability in school and trying to learn it on my own!

Vashi,

The problem with the 1/101 approach is that not all the outcomes are equally likely. Consider a simpler example. Suppose you just flip the coin twice. There are three possible outcomes, in terms of the total number of heads: 0, 1, or 2. But they are not equally likely: P(0 heads) = P(2 heads) = 1/4 and p(1 head) = 1/2. Do you see why?

thanks a lot, I understand now