An experiment has three possible outcomes: A, B, and C. If P(A)= P(B)and P(C)= 2P(A), what is the probability of each?
Can anyone help me get started?
Consider that P(A) + P(B) + P(C) = 1.
What do you mean?
The sum of the probabilities of each outcome must equal 1. Therefore, you can find the probability of any outcome by algebra.
P(A) + P(B) + P(C) = 1
I'll start you off.
Let x = P(A).
"P(A)= P(B)and P(C)= 2P(A)"
Then x = P(B) and 2x = P(C).
x + x + 2x = 1
So say I got x=1/4 would I substitute it in for P(A through C)?
Yes. x = 1/4, so P(A) = x = 1/4. Similarly, you can find P(B) and P(C).
Alright I think I got it
P(A)= 1/4
P(B)= 2/5
P(C)= 5/14
RIght?
Really good work.
"P(A)= P(B)and P(C)= 2P(A)"
We found that P(A) = x = 1/4. Therefore, P(B) = P(A) = x = 1/4.
P(C) = 2P(A) = 2x = 1/2
You can see that P(A) + P(B) + P(C) = 1/4 + 1/4 + 1/2 = 1
Of course! To find the probabilities of A, B, and C, we can start by assigning a variable to one of the probabilities and expressing the other probabilities in terms of that variable.
Let's say the probability of A is x. Since P(A) = P(B), the probability of B is also x.
According to the given information, P(C) = 2P(A). Substituting the value of P(A) with x, we can say P(C) = 2x.
Now, to find the overall probability, we sum up the probabilities of all possible outcomes, which should equal 1.
P(A) + P(B) + P(C) = 1
Substituting x for P(A) and x for P(B), we have:
x + x + 2x = 1
Simplifying the equation, we get:
4x = 1
To find x, we divide both sides by 4:
x = 1/4
So the probability of A (P(A)) and B (P(B)) is 1/4, and the probability of C (P(C)) is 2 times that:
P(A) = 1/4
P(B) = 1/4
P(C) = 2/4 = 1/2
Therefore, the probabilities are:
P(A) = 1/4
P(B) = 1/4
P(C) = 1/2