Probabilty of one tail followed by three heads with a biased coin

[battery88]

Homework Statement

A coin is biased so that the probability of heads is 2/3. What is the probability that exactly four heads come up when the coin is flipped seven times, assuming that the flips are independent?

Homework Equations

The Attempt at a Solution

C(7,4)(2/3)^4(1/3)^3 = (35*16)/3^7 = 560/2187

Now, I know this answer is correct, but what if we were asked to find probability of exactly one tail followed by three heads rather than exactly four heads: How would I calculate the answer for that?

Thank you.

 

[Ray Vickson]

Homework Statement

A coin is biased so that the probability of heads is 2/3. What is the probability that exactly four heads come up when the coin is flipped seven times, assuming that the flips are independent?

Homework Equations

The Attempt at a Solution

C(7,4)(2/3)^4(1/3)^3 = (35*16)/3^7 = 560/2187

Now, I know this answer is correct, but what if we were asked to find probability of exactly one tail followed by three heads rather than exactly four heads: How would I calculate the answer for that?

Thank you.

How would you calculate the probability of getting T on toss 1? If your first toss is T, does that affect the probability that the next three tosses all give H?

 

[battery88]

No, it doesn't since each toss is independent. So, it would be calculated the same way?

 

[haruspex]

Now, I know this answer is correct, but what if we were asked to find probability of exactly one tail followed by three heads rather than exactly four heads: How would I calculate the answer for that?

The four heads were out of 7 tosses. Are your 1T3H out of just four tosses? If not, what do you mean?

 

[bossman27]

The four heads were out of 7 tosses. Are your 1T3H out of just four tosses? If not, what do you mean?

I believe he was referencing another problem he posted on PF. But yeah, I assume that THHH would be out of four tosses; obviously any later ones would be superfluous.