Hello everyone,
I think there might be a mistake in the “Given Probability” video.
Around the 2-minute mark, Greg explains that we can use the number of possible cases to determine the probability. However, in this particular example, that only works because the probability is 50%, I think. Using this method with a different probability—for instance, 75%—would still yield the same result of 4/11, which would be incorrect.
You can also see this issue in the rain example at the end. Using the correct method, the probability is 1/33. But if you only consider the number of possible cases (there are 11 combinations with at least two rainy days, and 1 of them has exactly four rainy days), you’d get 1/11 instead.
Am I correct in my reasoning, or did I make a mistake?
Thanks in advance!
Can you post a reference image?
Hi!
It’s another example but shows the problem quite well. Here again, it only works due to the probability of 50% and wouldn’t work for any other probabilities.
You’re correct.
It becomes more clearer when you actually consider the definition of a conditional probability.
P(R = 4| R \geq 2) = \frac{P(R=4 \, \cap \, R \geq 2)}{P(R \geq 2)} = \frac{P(R = 4)}{P(R = 2) + P(R = 3) + P(R = 4)} = \frac{{\color{red}1} \left( \frac 13\right)^4}{{\color{red}6} \left( \frac 13\right)^2\left( \frac 23\right)^2 + {\color{red}4} \left( \frac 13\right)^3 \left( \frac 23\right) + {\color{red}1} \left( \frac 13\right)^4}
Note that the red numbers (the number of arrangements) are multiplied by their corresponding probabilities. You can’t just ignore the corresponding probabilities unless everything is equiprobable, as you noted.
FYI: This is the “rain” question from prepswift that you brought up.
