Problem Solving

A fair coin is flipped 3 times, what is the probability it will land heads up exactly twice?

A .125
B .25
C .375
D .50
E .666

Thanks!

lateraldeath wrote:A fair coin is flipped 3 times, what is the probability it will land heads up exactly twice?

A .125
B .25
C .375
D .50
E .666

Thanks!

Two of the easiest ways to answer coin flip questions are using a tree diagram and the coin flip formula.

I've attached the diagram as a document.

To read the diagram, start with either the first heads or tails and then go across to the right (pretend there are lines between the entries in the 1st and 2nd columns, and those in the second and third columns).

For example, going across the top, one possible string of outcomes is h-h-h; another is h-h-t... and so on.

So, if we want to know the probability of exactly 2 heads, we simply count the number of branches with exactly 2 heads on them and the total number of branches.

There are 3 that match what we want: h-h-t, h-t-h and t-h-h. There are 8 total branches, so the probability of exactly 3 heads is 3/8 or .375.

The diagram is great for 3 flip questions, since it's easy to draw and read.

The other method is to use the coin flip formula. Let's start with the basic probability formula:

probability = (# of desired outcomes)/(total # of possibilities)

For a coin flip question, the total # of possibilities is always 2^n, where n is the number of flips, since each new flip doubles the total branches of our tree.

(As an aside, the numerator of the answer to a coin flip question will ALWAYS be a power of 2, i.e. 1, 2, 4, 8, 16, 32, ..., so you can eliminate any answer choice that doesn't match - on this question, we can quickly eliminate .666 = 2/3.)

So, for 3 flips, the total # of possibilities is 2^3 = 8.

The # of desired outcomes is the number of different subgroups of 2 heads we can make out of 3 flips. Subgroups! You know what that means - combinations. So, the numerator of our formula is nCk, where k is the number of required results.

Let's apply the formula to the question you posted:

What's the probability of getting exactly 2 heads if you flip a fair coin 3 times?

n = 3; k = 2

Prob = nCk/2^n = 3C2/2^3 = (3!/2!(3-2)!)/8 = (3*2/2)/8 = 3/8 = .375

A fair coin is flipped 3 times, what is the probability it will land heads up exactly twice?

Ans) C .375


heads show up exactly twice when a coin is flipped 3 times is

HHT
HTH
THH
There fore, you can expect exactly 2 heads show up = 3

Total out comes = 8

HHH
HHT
HTH
THH
TTH
THT
HTT
TTT

Probability = possible out comes/ Total outcomes
probability = 3/8 = .375

Great post by Stuart but i think mistakenly at one place Stuart has mentioned numerator (As an aside, the numerator of the answer to a coin flip question will ALWAYS be a power of 2, i.e. 1, 2, 4, 8, 16, 32, ..., so you can eliminate any answer choice that doesn't match - on this question, we can quickly eliminate .666 = 2/3.) in place of denominator. It may be corrected.