Question

Three fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X.

Answer 1

Hint: Let us find the total number of outcomes in which the head comes 0 times, 1 times, 2 times or 3 times.

Complete step-by-step answer:
As we all know that coin is fair and tossed three times.
So, the maximum number of heads that can occur will be 3.
And the set of possible outcomes after three tosses will be { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }.
Where H denotes that head is occurred and T denotes that tail is occurred.
So, the total number of outcomes is 8.
As it is given that X denotes the number of heads.
So, X can be 0, 1, 2 or 3.
As we know that according to probability formula probability of a getting a favourable outcome = $${\displaystyle \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}}$$.
So, if (X = 0) (no head occurred) possible outcomes will be { TTT }.
So, P (X = 0) = $${\displaystyle \frac{1}{8}}$$
If (X = 1) (head occurred once) possible outcomes will be { HTT, THT, TTH }.
So, P (X = 1) = $${\displaystyle \frac{3}{8}}$$
If (X = 2) (head occurred twice) possible outcomes will be { HHT, HTH, THH }.
So, P (X = 2) = $${\displaystyle \frac{3}{8}}$$
If (X = 3) (all three are heads) possible outcomes will be { HHH }.
So, P (X = 3) = $${\displaystyle \frac{1}{8}}$$
Hence, the probability distribution of X is shown in the table below.

Note: Whenever we come up with this type of problem then first, we divide it into all possible cases like when (X = 0, X = 1, X = 2, X = 3). And then we find the probability of all the cases using probability formula which states that probability formula probability of a getting a favourable outcome = $${\displaystyle \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}}$$, and then plot them into a table.