Question

A coin is tossed three times. Find the probability of the following events.
(1) A: getting at least two heads
(2) B: getting exactly two heads.
(3) C: getting at most one head.

Answer 1

Hint: A coin is tossed three times, so the sample space:$\{HHH,HTH,THH,TTH,HHT,HTT,THT,TTT\}$, so, total number of possible outcomes = 8. Use the formula, $P(event)={\displaystyle \frac{No.ofFavourableOutcomes}{TotalNo.ofPossibleOutcomes}}$, where number of favourable outcomes means the outcomes which may take place for that event and total number of possible outcomes means the total possibilities that may occur when a coin is tossed three times. According to given cases find the number of favourable outcomes and probability of that particular case.

Complete step-by-step answer:
(1) A: getting at least two heads
 $P(A)=P(GettingTwoHeads)+P(GettingThreeHeads)$
Now for getting two heads, the number of favourable outcomes will be 3 which are $\{HTH,THH,HHT\}$.
And the total no. of possible outcomes is 8 which are $\{HHH,HTH,THH,TTH,HHT,HTT,THT,TTT\}$.
Now for getting three heads, no. of favourable outcomes will be 1 which is $\{HHH\}$.
Therefore, probability is-
 $P(A)={\displaystyle \frac{3}{8}}+{\displaystyle \frac{1}{8}}={\displaystyle \frac{4}{8}}={\displaystyle \frac{1}{2}}$ (using formula $P(event)={\displaystyle \frac{No.ofFavourableOutcomes}{TotalNo.ofPossibleOutcomes}}$)
$\therefore P(A)={\displaystyle \frac{1}{2}}$

(2) B: getting exactly two heads
Now for getting two heads, no. of favourable outcomes will be 3 which are $\{HTH,THH,HHT\}$.
And the total no. of possible outcomes is 8 which are $\{HHH,HTH,THH,TTH,HHT,HTT,THT,TTT\}$.
Therefore, the probability is-
$$\begin{gathered} P(B)=P(GettingExactlyTwoHeads) \\ P(B)={\displaystyle \frac{3}{8}} \end{gathered}$$ (using formula $P(event)={\displaystyle \frac{No.ofFavourableOutcomes}{TotalNo.ofPossibleOutcomes}}$)

(3) C: getting at most one head.
Getting atmost one head means number of heads can be 0 or 1 i.e getting no head and one head

Now for getting no heads, no. of favourable outcomes will be 1 which is $\{TTT\}$.
And the total no. of possible outcomes is 8 which are $\{HHH,HTH,THH,TTH,HHT,HTT,THT,TTT\}$.
Now for getting one head, no. of favourable outcomes will be 4 which is $\{HTT,THT,TTT\}$.
Therefore, probability is-
 (using formula $P(event)={\displaystyle \frac{No.ofFavourableOutcomes}{TotalNo.ofPossibleOutcomes}}$)
$$\begin{gathered} P(C)=P(GettingNoHeads)+P(GettingOneHeads) \\ P(C)={\displaystyle \frac{1}{8}}+{\displaystyle \frac{3}{8}}={\displaystyle \frac{4}{8}}={\displaystyle \frac{1}{2}} \\ \therefore P(C)={\displaystyle \frac{1}{2}} \end{gathered}$$

Note- Whenever such a type of question appears note down all the outcomes of the event and the possible outcomes in the particular given case, as given in question the coin is tossed 3 times. And then find the probability of all the cases using the standard formula.