Like. I know they are different and that one is less likely. But could someone explain this in a way that I can explain it to my partner? I know it is true but cannot remember anything about why and how to explain my point.
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The odds of flipping at least 1 head in 4 flips is about a 94% (93.75 to be exact) chance. So you might think that every flip has a 94/4 or 23.5% chance of coming up heads, but we know that’s not the case, so what’s going on?
The first flip gives us a 50% chance of giving an immediate heads. If that’s the case, we can stop after one flip, we got our heads. Only half of the time we will need a second flip.
During the second flip, half of the time, we can expect to get our heads. So half of the time we need a 2nd flip and half of half of the time (25%) we will get our heads and be done after the second flip. So far, we have a 50% chance of getting heads on the first flip and a 25% chance of getting heads on the second flip. (Remember that’s the 50% odds of getting a heads times the 50% odds that we even need to flip the 2nd coin). That’s 50+25=75% leaving 25% of the outcomes that will require a 3rd flip.
This logic continues. 25% of the time we need a 3rd flip and half of those outcomes (half of 25 is 12.5%) we get the heads and are done. So 50+25+12.5 = 87.5% of the time, we get our heads in one of the first 3 flips, only 12.5% of attempts will we need to go to the 4th flip.
Of those 12.5% of attempts that require the 4th flip, half of them will give us the heads we need (half of 12.5 is 6.25%) and the other half (the remaining 6.25%) we will fail our 4th flip as well because we flipped 4 tails in a row. What are the odds of getting 4 tails out of 4 flips? It’s 1/2⁴ because there are 2⁴ possible outcomes and only one of them is TTTT. What’s 1/2⁴? It’s 6.25% awesome, we verified our logic.
And what are the odds of getting at least one heads? 50% from the 1st flip plus 25% from the 2nd, plus 12.5% from the 3rd and plus 6.25% from the 4th gives us a grand total of 93.75% what’s more, the odds that we don’t get any heads is 6.25% and since the only possible outcomes from 4 flips is at least one heads or no heads, we should get 100% if we add up the odds of both possibilities: 93.75+6.25=100 great! More verification we did it right.
If you just take the 50+50+50+50=200% chance of each flip that is double counting. Because if the second flip comes up heads, half of the time, the first flip was already heads, so we already counted that probability once. We don’t need to count it again.
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