Question

How many possible outcomes if the coin is tossed:
(i) Four times
(ii) Five times
(iii) n times?

Answer 1

Hint:When tossing a coin, there are 2 outcomes, Head (H) and Tail (T). Each toss is independent of the other. Write the possible outcome with H and T when tossing a coin 4 times. Using the multiplication theorem of probability for finding outcome when tossing a coin 5 and n times.

Complete step-by-step answer:
(i) Every time we toss a coin, there are two possibilities: Head (H) or Tail (T). As one coin is tossed four times, each toss is independent of the other.
Let us note down the possible outcomes of tossing a coin 4 times
{TTTT ,TTTH, TTHT, THTT, HTTT ,TTHH, THTH, HTHT, HHTT, THHT, HTTH,THHH, HTHH, HHTH, HHHT,HHHH}
Thus we can say that there are 16 possible outcomes, while tossing a coin 4 times.

(ii) Now, a coin is tossed five times, each toss is independent of the other. By multiplication theorem, we can say that if A and B are two independent events, then the probability that both will occur is equal to the product of their individual probabilities.
So when we toss a coin 5 times, we first get a head or Tail and again on the second toss we get a H or T. Similarly, it goes on. Thus we can use the multiplication theorem, as per the fundamental theorem of counting. As there events occur one after the other, by multiplication theorem, we can say that the sample space produced will be equal to,
Sample space = \[2\times 2\times 2\times 2\times 2={{2}^{5}}=32\]
\[\therefore \] Total possible outcome of tossing a coin 5 times = 32.

(iii) Possible outcome of tossing a coin n times.
By multiplication theorem, we can write that, when a coin is tossed one we get head or tail. Similarly, tossing a coin n times means, the sample space will be,
Sample space = \[2\times 2\times 2\times .........\times 2={{2}^{n}}\]
\[\therefore \] Total possible outcome of tossing a coin n times = \[{{2}^{n}}\].
\[\therefore \] Possible outcome of tossing a coin 4 times = 16
Possible outcome of tossing a coin 5 times = 32
Possible outcome of tossing a coin n times = \[{{2}^{n}}\]

Note: You can use \[{{n}^{r}}\] to calculate this where, n is the number of options you have each step and r is the number of trials. For a coin there are 2 possible outcomes, thus n = 2. Now the coin is tossed 5 times, thus r = 5.
\[\therefore \] Possible outcome of tossing a coin 5 times = \[{{n}^{r}}={{2}^{5}}=32\]
Similarly possible outcome of tossing a coin n times \[{{n}^{r}}={{2}^{n}}\].