# unfair coin flip probability calculation

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.

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.

• Thanks Ferdi, vote up for your reply. But I mean flip 6 times, other than flip 3 times as you mentioned in your reply. And I do not care the last 3 time result. Maybe you can read my question again. Let me know if anything unclear. :) – Lin Ma Oct 5 '16 at 6:10
• Thanks Ferdi, two more comments, I think I mean for the first time times, it is HTT, THT or TTH, the last 3 times does not mater (total 6 times flip), so I think it should be 0.3 * 0.7 * 0.3 * 0.3 ^ 3 other than 0.3 ^ 3 * 0.3 * 0.7 * 0.3? The other comment is, not sure why you are using 0.3^3, the last 3 times could be any combinations of head and tail. – Lin Ma Oct 5 '16 at 6:28
• Thank you for correcting me. I edited my post once again. I just stood up, so I am still tired and not on the peak of my daily performance. – Ferdi Oct 5 '16 at 6:32
• A further question, suppose I still flip 6 times, and I do not care first 3 flip results, and only care the last 3 flips results, for the 3 events, they are also of the same probability -- ???HTT, ???THT and ???TTH? And their probability is the same as HTT???, THT??? and TTH??? (where I care first 3 flip results, and do not quite care last time flips)? – Lin Ma Oct 5 '16 at 7:09
• Hi Ferdi, if you could comment on my above question, it will be great. Thanks. – Lin Ma Oct 5 '16 at 19:23