Question

How many possible combination outcomes consist of at least three tails when you toss a fair coin four times?

Answer 1

Here are all the possible outcomes
when you toss a coin 4 times.

Arrows point to the outcomes that
include at least 3 tails.

H H H H
H H H T
H H T H
H H T T
H T H H
H T H T
H T T H
H T T T  <===
T H H H
T H H T
T H T H
T H T T    <===
T T H H
T T H T    <===
T T T H    <===
T T T T    <===

There are five of them.

Answer 2

The total number of possible combinations of outcomes that consist of at least three tails when you toss a fair coin four times is $\boxed{5}$.

Explanation:

Combination is joining or combining different things in which each element is individually distinct. If ‘r’ things are to be selected from n different things, then the total different combinations are formulated as shown below.

$\begin{aligned}\text{Total different combinations} &= C(n,r)\\&= \dfrac{n!}{(r!)(n-r)!}\end{aligned}$

Getting heads or tails on a coin is the same as picking or not picking an item for the combination.

Now, consider that the picking up of an item in the combination is same as getting tails and the chances of not picking up an item as getting heads.

The fair coin is tossed $4$ times so $n$ equals $4$. The outcomes should consist of at least three tails that is, it can contain either $3$ or $4$ tails to get considered as a favorable outcome.

$\boxed{\begin{aligned}\text{Total possible outcomes} &= C(4,3) + C(4,4)\\& =\frac{4!}{3!1!}+\frac{4!}{4!}\\&=4 + 1 \\&= 5\end{aligned}}$

Therefore, there are a total of five different combinations of outcomes of four coins such that the outcomes consist of at least three tails.

 

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Answer Details:

Grade: Middle School

Subject: Mathematics

Chapter: Permutation and combination

Keywords: combination, coin, heads, tails, outcomes, toss, chances, arrangement, selection, at least, at most, three, four, total outcomes.