Can someone help me understand these types of probability problems?

[super112]

You have four coins, to win you need 3H and 1T. Calculate your probability of winning.

So I always do (1/2)^3 * (1/2) = 1/16

but answer says I need to multiply by 4. I don't know if it's just me, but sometimes I feel like the problem expects me to multiply the end probability fraction and sometimes it doesn't. So I'm hoping for some clarification on when I'm supposed to multiply the probability (aka (1/16) *4)

Thanks

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[FeralisExtremum]

By doing (1/2)^3 * (1/2), you are calculating the chance of flipping 3 H's and then 1 T on four coins. This comes in handy later on.

The problem is that you are being asked the probability of winning, not the probability of flipping 3 H's and then 1 T in that specific order. The question did not ask "what is the chance of flipping H on the first coin, H on the second, H on the third, and T on the fourth?" You could win any of the following ways:

T H H H
H T H H
H H T H
H H H T

Since these are all winning arrangements, you take your chance of flipping that outcome (you did this in your first step, 1/16) and you multiply it by the number of different arrangements you could get that outcome from (the 4 I outlined above). (1/16) * 4 = 4/16 or 1/4.

The key is usually paying attention to whether you are being asked to find a specific outcome (e.g. flipping an H and then a T and then an H, a specific outcome that can only happen one way - which is just multiplying fractions) or whether you are being asked to find a an outcome that can happen multiple ways (flipping an H, T, and H in any sequence - which would involve multiplying the probability fractions times the number of desired sequences).

 

[DAT Destroyer]

You have four coins, to win you need 3H and 1T. Calculate your probability of winning.

So I always do (1/2)^3 * (1/2) = 1/16

but answer says I need to multiply by 4. I don't know if it's just me, but sometimes I feel like the problem expects me to multiply the end probability fraction and sometimes it doesn't. So I'm hoping for some clarification on when I'm supposed to multiply the probability (aka (1/16) *4)

Thanks

The reason we multiply here by 4 is because there are 4 possibilities to get 3 H and 1 T. ( HHHT HHTH HTHH THHH). If you look at it as an arrangement problem we have 4!/3! = 4 ways to arrange 4 things where 3 things are similar.

 

[super112]

Thanks guys, makes a lot of sense now. But here's a side question, is there a quick and easy way to see how many possibilities there are? For example what if they say 10 coins and to win you need 6H and 4T?

 

[FeralisExtremum]

Thanks guys, makes a lot of sense now. But here's a side question, is there a quick and easy way to see how many possibilities there are? For example what if they say 10 coins and to win you need 6H and 4T?

This is what you would use the combination/permutation formulas for. For example, if you need exactly 6 H's and 4 T's to win and want to know the probability, you would use the combinatorial formula:

Where n is the number of things to choose from and r is the number of desired events (we're going to use 6 here, as we want exactly 6 heads)

Chance to flip specific set: (1/2)^10

# of winning sets: n C r or in this case 10 C 6:
10!/ [ 6! (10-6)!] = 10! / [6! 4!] = 210 ways

Multiply chances to flip specific set * # of possible winning sets: [(1/2)^ 10] * 210 = 210/1024 = 105/512

But again, you will need to know when to use the permutation formula vs the combination formula. If you need to brush up on this stuff I highly recommend Khan Academy's math videos.

 

[super112]

Alright! I never even thought about using the combination equation. And just to check, we could have used either 6 or 4 for r correct? Because getting 6H automatically means other 4 are tails and vice versa?

 

[FeralisExtremum]

In this case yes, either value for r would get you the same answer.

 

[super112]

Thanks for your help!! I really appreciate it!