Question:
Fifteen coupons are numbered 1, 2....... 15, respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9, is
Fifteen coupons are numbered 1, 2....... 15, respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9, is
Updated On: Jun 14, 2022
- $\bigg(\frac{9}{16}\bigg)^6$
- $\bigg(\frac{8}{15}\bigg)^7$
- $\bigg(\frac{3}{5}\bigg)^7$
- None of these
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The Correct Option is D
Solution and Explanation
Since, there are 15 possible cases for selecting a coupon and seven coupons are selected, the total number of cases of selecting seven coupons = 15$^7$
It is given that the maximum number on the selected coupon is 9, therefore the selection is to be made from the coupons numbered 1 to 9.
This can be made in 97 ways. Out of these 9$^7$ cases, 8$^7$ does not contain the number 9.
Thus, the favourable number of cases = 9$^7$ - 8$^7$.
$\therefore \, \, $ Required probability $=\frac{9^7 -8^7}{15^7}$
It is given that the maximum number on the selected coupon is 9, therefore the selection is to be made from the coupons numbered 1 to 9.
This can be made in 97 ways. Out of these 9$^7$ cases, 8$^7$ does not contain the number 9.
Thus, the favourable number of cases = 9$^7$ - 8$^7$.
$\therefore \, \, $ Required probability $=\frac{9^7 -8^7}{15^7}$
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Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.