What are the odds (coin flipping question)?

Factual Questions
1

Let’s say I have 3 quarters and I flip each quarter separately. What are the odds that all 3 quarters would have landed heads up, or that all 3 quarters would have landed tails up (no mix of heads and tails)? How would you figure that out?

2

50% of 50% of 50% is 12.5%…

3

Easy enough even I know that one. Out of the eight possibilities – you can write down all eight – HHH, HHT, HTH, HTT, THH, THT, TTH, TTT – one is TTT and one is HHH. One chance in eight for HHH, one chance in eight for TTT.

4

Not quite. It’s three of either. So the first flip can be anything. Only the second and third flips are then contstrained to being the same as the first.

So it’s 50% x 50% = 25%.

5

Well, yeah; 12.5% chance of all-heads, 12.5% chance of all-tails; I figured it was implied that adding those together would get 25%.

6

If you want a reasonably simple method to calculate other heads/tails results:

If you flip n coins, expand (H+T)^n. If you want to know the probability of x heads and n-x tails, look for the H^x T^(n-x) in your expansion. Isolate it, plug in your probabilities for both H and T (50% of each if they are fair coins (and by the way, all the coins need to have the same bias for this method to work: For example if one coin is biased 70% H, 30% T, then all coins must have that same probability)) and the result is the probability you want.

For example, with three coins: (H+T)^3 = H^3 + 3H^2 T + 3H T^2 + T^3. Assuming the coins are all fair, the probability of flipping 2 heads, 1 tail would be 3H^2 T = 3(.5)^2 (.5) = 37.5%.

7

All the responses so far have answered as if the question were “What is the probability?”

But someone who is careful to distinguish between probability and odds might say that the odds are 1 to 7 of all three quarters coming up heads (and 2 to 6, or 1 to 3, of all three coming up the same, either heads or tails). By this interpretation, “odds” means the ratio of favorable to unfavorable outcomes.

8

It’s also worth pointing out there’s no difference between flipping 3 coins once each, or one coin 3 times. Or for that matter one coin twice and a second coin once.

At least assuming perfectly fair coins.