Question

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?

Answer 1

C(7, 4)(2/3)^4(1/3)^3 is the answer.
Could someone explain how they reached this conclusion?

Answer 2

The binomial distribution applies in this case. Have you studied the binomial distribution?

Answer 3

It does!
Thanks, I got it now!

Answer 4

On a side note, could you help me with this?

Answer 5

Given a coin that has a head 3 times more likely to come up than a tail, what's the probability of getting a head and a tail?

Answer 6

I think the solution is (3/4) (1/4) + (1/4) (3/4) = 6/16 , but I'm not sure.

Answer 7

This question is not clear. However if it is asking for the probability of getting a head first and then a tail on two successive tosses, then the calculation would be as follows:
On the first toss, P(head) is 3/4.
On the second toss, P(tail) is 1/4.
The events 'head on first toss' and 'tail on second toss' are independent, meaning the outcomes have no effect on each other.
Therefore P(head on first toss, tail on second toss) = 3/4 * 1/4 = 3/16

Answer 8

I agree, and I think that's true.

Answer 9

However if the order in which a head and a tail were obtained does not matter, the result would be as you wrote it.

Answer 10

If one was to ask the probability of getting two head when tossing two of these coins, would you add 3/4 + 3/4 ?

Answer 11

No. The results of each toss are independent, therefore the probabilities of heads must be multiplied. This gives 3/4 * 3/4 = 9/16.