Probability of second heads on fifth flip

[schinb65]

Hello,

I am wondering how to do this problem. I have a way that gave me the correct answer but I would not want to do this every time or for large numbers.

What I did:
So 2^5 equals 32 since the probability is 1/2 and I want 5 flips. Then I have 4 ways that the 2nd head will land on the 5th flip. So I have 4/32. which is 12.5%

I also tried Binomial distribution but did not get the correct answer.
5C2(.5)^2(.5)^3. I know that this will not work since the second head can be in any location but do not remember how I can make it work for this problem or if it would even work.

 

[Jameson]

Re: second head on fifth flip

Hi schinb65,

The problem is really important for this question because it gives us the situation from which we choose how to answer it. Flipping coins is usually binomial related but it's better to be too clear than not clear enough. I believe the question is something like "if you flip a coin until you reach a total of 2 heads what is the probability that the 2nd heads occurs on flip number 5?"

This appears to use the negative binomial distribution. I won't define everything here but there are two ways of thinking of this distribution: 1) counting the number of failures before the last success 2) counting the number of trials to achieve k successes

You have three variables to use in the probability function: k, r and p. k is the number of successes, r is the number of failures and p is the probability of a success.

If \(\displaystyle P(X=k)=\binom{k+r-1}{k}(1-p)^r p^k\) can you use the above information to solve this? I think the first try you made is very close the to answer. The binomial coefficient should be different though. It can't be \(\displaystyle \binom{5}{2}\) because that would allow the 5th flip to be tails some of the time, which isn't allowed.

Jameson

 

[soroban]

Re: second head on fifth flip

Hello, schinb65!

A coin is tossed 5 times.
Find the probability that the second Head occurs on the fifth toss.

In the first 4 tosses, we want 3 Tails and 1 Head in some order.
. . This probability is: .$$(_4C_3)\left(\tfrac{1}{2}\right)^3\left(\tfrac{1}{2}\right) \:=\:\tfrac{1}{4}$$

Then we want a Head on the fifth toss: $$\tfrac{1}{2}$$

Therefore, the probability is: .$$\tfrac{1}{4}\cdot\tfrac{1}{2} \;=\;\tfrac{1}{8}$$

 

[Jameson]

Re: second head on fifth flip

soroban just owned me, haha. (Punch)

This particular question can be answered like soroban showed, and the method is pretty intuitive. Like I mentioned, the problem the OP had was with understanding which terms are being arranged and which are not.

If these questions stay simple enough that you can imagine the experiment in your head, then that way is fine. If you are asked to write the explicit probability formula for some situation though and it's expected to be able to shift between writing in terms of counting the failures versus counting the successes, then you will probably be looking into the fine print but it is a bit overkill for this question I suppose.

 

[CellCoree]

Hello,

Thank you for sharing your approach to solving this problem. It seems that you have correctly identified the probability of getting a second heads on the fifth flip as 4/32 or 12.5%. However, I understand your concern about this method being time-consuming for larger numbers.

One way to approach this problem is by using the concept of conditional probability. This means that we can calculate the probability of getting a second heads on the fifth flip, given that we have already gotten one heads in the first four flips.

So, let's break down the probability into two parts: the probability of getting one heads in the first four flips and the probability of getting a second heads on the fifth flip.

The probability of getting one heads in the first four flips can be calculated as (1/2)^4 = 1/16. This means that out of all the possible combinations of four flips (which is 2^4 = 16), only one will have one heads.

Now, we only need to consider the remaining two flips. The probability of getting a second heads on the fifth flip is 1/2. This is because at this point, we have already gotten one heads and we are only looking at the probability of getting a second heads on the fifth flip.

Therefore, the final probability can be calculated as (1/16) * (1/2) = 1/32 or approximately 3.125%.

I hope this helps. Remember, there are multiple ways to approach a problem and it's always good to explore different methods to find the most efficient one. Keep up the good work!