Question

One Indian and four American men and their wives are to be seated randomly around a circle table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is
1). $\dfrac{1}{2}$
2). $\dfrac{1}{3}$
3). $\dfrac{2}{5}$
4). $\dfrac{1}{5}$

Answer 1

To solve this problem, first we need to understand the concept of probability and after that try to find out the probability of each couple seated randomly around a circle and then find out the probability of event when each American man is seated adjacent to his wife and after that use the formula of conditional probability and you will get your required answer.

Complete step-by-step solution:
Probability is defined аs the роssibility оf an event to оссur. For example: if we toss a coin then there are only two possibilities: either head will occur or tail. The probability of occurrence of head is half and the probability of occurrence of tail is half.
The formula for Probability is given аs the ratio of the number оf fаvоrаble events to the total number of possible outcomes. There are three types of probability:
Theoretical Probability
Experimental Probability
Axiomatic Probability
In theoretical Probability, it depends on all the possible chances of something to happen.
In experimental probability, it depends on the observations of the experiment. It is calculated as the number of all the possible outcomes to the number of trials.
In axiomatic probability, it tells the chances of the occurrence or non-occurrence of the event.
Now, according to the given question:
Let $A$ be the event when Indian man is seated adjacent to his wife
And $E$ be the event when each American man is seated adjacent to his wife.
Each couple is taken together as one entity, so we have total $5$ things to be arranged in a circle, which can be done in $(5-1)!=4!$ number of ways. Also, each couple as one entity and can be seated in $2$ ways.
So, $n(A\cap E)=4!{{(2!)}^{5}}$
Each American couple is taken as one entity, and an Indian man and woman separately. Hence, the arrangement of $6!$ things in a circle is $5!$ . Also each American couple as one entity can be seated in $2$ ways.
So, $n(E)=5!{{(2!)}^{4}}$
Now, the required probability is given as: $P(A|E)=\dfrac{n(A\cap E)}{n(E)}$
$\Rightarrow P(A|E)=\dfrac{4!{{(2!)}^{5}}}{5!{{(2!)}^{4}}}$
$\Rightarrow P(A|E)=\dfrac{2}{5}$
Hence, the correct option from all the above options is $3$.

Note: Uses of probability in our day to day life: Use to predict the weather changes, Used in games such as poker, blackjack etc to know the probability of the person to win, Political analysts use the probability to predict the outcomes of the election’s result, Winning or losing lottery is also an interesting example of it etc.