Suppose I have an unfair coin, and the probability of flip a head (H) is p, probability of flip a tail (T) is (1-p).
If I flip the coin 6 times, wondering if the probability of HTT???, and the probability of THT???, and the probability of TTH??? are the same? Suppose each flip is independent. ? means do not care if head or tail. Thanks.
I calculated they are the same, ask here to get advice from expert if my calculation is correct.
Best Answer
Yes, all the three events are independent. The cumulative law gives you:
t * h *t = h * t * t = t* t* h
If you enter number you will get:
0.3 * 0.7 * 0.3 = 0.7 * 0.3 * 0.3 = 0.3 * 0.3 * 0.7 = 0.063
Edit: It does not matter what the first three times are because of the law of conditional probability.
P(A $\bigcap$ B) = P(A) * P(B) $\Leftrightarrow$ P(A) = $\frac{P(A)P(B)} {P(B)}$ = $\frac{P(A)\bigcap P(B)} {P(B)}$ = P(A|B)
You can also take an example. All equations have the same product.
0.3 * 0.7 * 0.3 *0.3 ^ 3 = 0.001701
0.7 * 0.3 * 0.3 *0.3^3= 0.001701
0.3 * 0.3 * 0.7 * 0.3^3= 0.001701
The last could be any combination of head and tail. 0.3^3 is just an example, but it works out equally if one or two or three of the variables is 0.7.