Consider a biased coin that comes up heads with probability 1/3 and tails with probability 2/3
Answers
This is a nice and easy one to do just by running the cases. This would be my starting table:
Option 1: Head, Head, Head
Option 2: Head, Head, Tail
Option 3: Head, Tail, Head
Option 4: Head, Tail, Tail
Option 5: Tail, Head, Head
Option 6: Tail, Head, Tail
Option 7: Tail, Tail, Head
Option 8: Tail, Tail, Tail.
As you may quickly observe: there are 3 throws, each with 2 options. This leads to 2 ^ 3 cases. This formula generalizes: n ‘throws’ with m ‘options’ leads to m ^ n different cases.
Now lets plug in some numbers. Heads come up in 2:3, therefor Tails comes up 1:3. This formula also generalizes. If you know all options but one, the probability for the last one is alway ‘1 - sum(other options)’:
Option 1: 2/3, 2/3, 2/3
Option 2: 2/3, 2/3, 1/3
Option 3: 2/3, 1/3, 2/3
Option 4: 2/3, 1/3, 1/3
Option 5: 1/3, 2/3, 2/3
Option 6: 1/3, 2/3, 1/3
Option 7: 1/3, 1/3, 2/3
Option 8: 1/3, 1/3, 1/3
So far so good! We now know the probability for each throw in each option. But what is the total probability for the entire option? The rules of probability state that for that you have to multiply the different probabilities for the throws. So here goes:
Option 1: 8/27
Option 2: 4/27
Option 3: 4/27
Option 4: 2/27
Option 5: 4/27
Option 6: 2/27
Option 7: 2/27
Option 8: 1/27
To check the result, you can add up the probabilities of the options. You’ll quickly notice that they add up to 27/27 = 1. Which indicates that indeed we covered all the cases.
Finally, let’s select the correct options we wanted. Turns out only options 2, 3 and 5 meet the criteria of having 2 heads. The total probability of those options is 12/27.
So that’s the answer you were looking for.
Note 1: there are fancy formulas for achieving this result. However, with so few options (only 8), running the cases seemed more instructive, showing the basic tricks of probability.
Note 2: You could also use percentages. I choose not to do that, because that would convolute the logic and would be inprecise. The fraction 12/27 is the exact probability.
hope it is helpfull
mark it as a brainliest answer