Question:
A coin is tossed $7$ times. Each time a man calls head. Find the probability that he wins the toss on more occasions.
A coin is tossed $7$ times. Each time a man calls head. Find the probability that he wins the toss on more occasions.
Updated On: Sep 3, 2024
- $\frac{2}{3}$
- $\frac{1}{2}$
- $\frac{3}{4}$
- $\frac{1}{3}$
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The Correct Option is B
Solution and Explanation
The man has to win at least $4$ times.
$\therefore $ Reqd. probability =
$^{7}C_{4} \left(\frac{1}{2}\right)^{4} .\left(\frac{1}{2}\right)^{3} + ^{7}C_{5} \left(\frac{1}{2}\right)^{5} . \left(\frac{1}{2}\right)^{2} + ^{7}C_{6} \left(\frac{1}{2}\right)^{6} . \frac{1}{2} + ^{7}C_{7} \left(\frac{1}{2}\right)^{7} $
$ = \left(^{7}C_{4} + ^{7}C_{5} + ^{7}C_{6} + ^{7}C_{7}\right) . \frac{1}{2^{7}} = \frac{64}{2^{7}} = \frac{1}{2} $
$\therefore $ Reqd. probability =
$^{7}C_{4} \left(\frac{1}{2}\right)^{4} .\left(\frac{1}{2}\right)^{3} + ^{7}C_{5} \left(\frac{1}{2}\right)^{5} . \left(\frac{1}{2}\right)^{2} + ^{7}C_{6} \left(\frac{1}{2}\right)^{6} . \frac{1}{2} + ^{7}C_{7} \left(\frac{1}{2}\right)^{7} $
$ = \left(^{7}C_{4} + ^{7}C_{5} + ^{7}C_{6} + ^{7}C_{7}\right) . \frac{1}{2^{7}} = \frac{64}{2^{7}} = \frac{1}{2} $
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Concepts Used:
Event
A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space. The likelihood of occurrence of an event is known as probability. The probability of occurrence of any event lies between 0 and 1.
Thus, an event is a subset of the sample space, i.e., E is a subset of S.
There could be a lot of events associated with a given sample space. For any event to occur, the outcome of the experiment must be an element of the set of event E.
Probability of Occurrence of an Event
P(E) = Number of Favourable Outcomes/ Total Number of Outcomes
Types of Events in Probability:
Some of the important probability events are:
- Impossible and Sure Events
- Simple Events
- Compound Events
- Independent and Dependent Events
- Mutually Exclusive Events
- Exhaustive Events
- Complementary Events
- Events Associated with “OR”
- Events Associated with “AND”
- Event E1 but not E2