Probability of coins

Question

If you flip 4 coins( at the same time ), write the sample space and answer the following questions:

a. What is the probability of all 4 coins landing on heads?

b. What is the probability of at least 3 coins landing on tails?

Answer 1

In order to answer probability questions like these, the sample space really is the right place to start, because once you can see every possible outcome the rest is just counting.

Each coin has 2 possibilities, heads or tails, and there are 4 coins, so the total number of outcomes is 2 x 2 x 2 x 2 = 16. Writing them all out, using H for heads and T for tails:

HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT

If you notice, listing them in a system like this, rather than randomly, is what keeps you from accidentally skipping one. I went through them as if I were counting up in binary with H as 0 and T as 1.

For part a, all four coins landing on heads is just the single outcome HHHH. One outcome out of sixteen:

P(all heads) = 1/16

For part b, "at least 3 tails" is the phrase to slow down on, because at least 3 means 3 tails OR 4 tails, not exactly 3. So we count both.

Exactly 3 tails means one coin came up heads and it could have been any of the four, which gives HTTT, THTT, TTHT and TTTH. That is 4 outcomes.

Exactly 4 tails is just TTTT, which is 1 outcome.

Adding those together gives 5 outcomes out of 16:

P(at least 3 tails) = 5/16

So, our final answers are 1/16 for part a and 5/16 for part b.

Answer 2

We are solving for the probability of all coins landing on heads (or tails), as long as they all land on the same side. The probability of one coin landing on a pre-chosen side is 1/2, so the probability of all four coins landing on the same side is (1/2)^4 = 1/16.

a. 1/2 x 1/2 x 1/2 x 1/2 = 1/16 or 0.0625 or 6.25%.

From the problem above (a), it is both the probability of landing all heads or all tails, which means that the answer 1/16 can be used for the next problem as well. 1/16 will represent P(4).

b. At least 3 coins on the same side means we must calculate at least 3, so 3 coins or 4 coins on the same side. P(4) is 1/16, and for P(3), we would use the binomial probability distribution since it consists of multiple combinations.

The equation for the number of combinations of getting x tails in n coin tosses:

n!

--------

x!(n-x)!

n = # of coin tosses

x = # of coins that land on tails

! is a factorial, for example 5! = 5 x 4 x 3 x 2 x 1

So

4! 4 x 3 x 2 x 1 24

--------- = ------------------ = -------------- = 4

3!(4-3)! 3x2x1 (4-3)! 6(1)

(1!) just equals 1

So there are 4 possibilities of the coins landing with exactly 3 tails within 16 total possible outcomes.

Therefore, 4/16 + 1/16 a.k.a. P(3) + P(4) {because it is AT LEAST 3 tails} = 5/16 or 0.3125 or 31.25% chance of at least 3 tails.

Answer 3

The sample space is asking what are the possible outcomes.

HHHH HHHT HHTH HHTT HTHH HTHT HTTT HTTH

THHH THHT THTH THTT TTHH TTHT TTTH TTTT

a) Probability can be calculated by the number of outcomes divided by the total number of possibilities.

P(4 heads) 1/16

b) P(at least 3 tails) = 4/16 = 1/4