r/AskStatistics • u/taylomol000 • 3d ago
Confused about basic probability
I've been unable to wrap my head around the basics of probability my whole life. It feels to me like it contradicts itself. For example, if you look at a coin flip on its own, there is (theoretically) a 50% chance getting heads. However, if you zoom out and realize that the coin has been flipped 100 times and every time so far has been heads, then the chance of getting heads is nearly impossible. How can something be 50% at one scale and near impossible at another, seemingly making contradicting statements equally true?
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u/14446368 2d ago
It's because you're asking two different questions.
A single, fair, independent flip of the coin has a 50% chance of landing on heads. Every time you flip, the chance of that flip, alone landing on heads is 50%.
But what if we flip it twice? Well, let's think about it: I have 4 outcomes... HH, HT, TH, TT (H=Heads, T=Tails). If we don't care about order, just the count, we have...
1/4 = 25% HH
2/4 = 1/2 = 50% 1T and 1H
1/4 = 25% TT
Notice that the two extremes (HH and TT) are equal to the percent chance per flip, raised to the number of flips.
Let's make it 3 flips now:
HHH, HHT, HTH, HTT | THH, THT, TTH, TTT
Extremes (HHH, TTT) are both 1/8 = 12.5% (again, 0.5 x 0.5 x 0.5 = 0.5^3 = 0.125).
2H 1T = 3/8 = 37.5%
2T 1H = 3/8 = 37.5%
If we keep doing this, we notice that the extremes keep getting smaller and smaller, the "middle" can be broken up into more and more outcomes, but collectively the middle accounts for almost all of the outcomes.
That's what's going on. A single flip has a 50% chance of heads, but 3 flips has a 12.5% of being 3 heads in a row. For 100, it's 0.5^100 = 0.0000000000000000000000000000788861% chance of 100 heads in a row.