Question

Odds 8 to 5 against a person who is 40 years old living till he is 70 and 4 to 3 against another person now $50$ till he will be living $80$. Probability that one of them will be alive next $30$ years
$A)\dfrac{{59}}{{91}}$
$B)\dfrac{{44}}{{91}}$
$C)\dfrac{{51}}{{91}}$
$D)\dfrac{{32}}{{91}}$

Answer 1

First, we need to know the concept of probability.

Probability is the term mathematically with events that occur, which is the number of favorable events that divides the total number of outcomes.
Since there are a total of two events which are person A and person B.
We will find separately the probability that A will die and B will die, but by the probability formula for both independent events, we will get the required result.

Formula used:
$P = \dfrac{F}{T}$where P is the overall probability, F is the possible favorable events and T is the total outcomes from the given.
$Ind = P(\overline A ).P(B) + P(A).P(\overline B )$ where $\overline A B,A\overline B$ are independent events

Complete step-by-step solution:
Since given that we have two events, which are $8$ to $5$ against a person who is $40$years old living till he is $70$ and $4$ to $3$ against another person now $50$ till he will be living $80$
Thus, the first event is Person A, and the difference is $30$ the years he has died. Thus, the favorable event is $8$ and the possible total outcome is $8 + 5$.
Hence, we get $P(A) = \dfrac{8}{{8 + 5}} \Rightarrow \dfrac{8}{{13}}$ (probability that A will die)
Thus, we also get $P(\overline A ) = 1 - P(A) = 1 - \dfrac{8}{{13}} \Rightarrow \dfrac{5}{{13}}$ (probability that A will living)
Similarly, we have the second event be Person B, and the difference is also ($50$ till he will be living $80$)$30$ years he has died. Thus, the favorable event is $4$ and the possible total outcome is $4 + 3$.
Hence, we get $P(B) = \dfrac{4}{{4 + 3}} \Rightarrow \dfrac{4}{7}$ (probability that B will die)
Thus, we also get $P(\overline B ) = 1 - P(B) = 1 - \dfrac{4}{7} \Rightarrow \dfrac{3}{7}$ (probability that B will be living)
Therefore, we need to find the probability that one of them will be alive next thirty years is given $Ind = P(\overline A ).P(B) + P(A).P(\overline B )$ where $\overline A B,A\overline B$ are independent events.
Thus, substituting the values, we get $Ind = P(\overline A ).P(B) + P(A).P(\overline B ) \Rightarrow \dfrac{5}{{13}}.\dfrac{4}{7} + \dfrac{8}{{13}}.\dfrac{3}{7}$
Further solving we get $P(\overline A ).P(B) + P(A).P(\overline B ) = \dfrac{5}{{13}}.\dfrac{4}{7} + \dfrac{8}{{13}}.\dfrac{3}{7} \Rightarrow \dfrac{{20}}{{91}} + \dfrac{{24}}{{91}}=\dfrac{{44}}{{91}}$
Hence, the option $B)\dfrac{{44}}{{91}}$ is correct.

Note: First, let us assume the overall total probability value is $1$ (this is the most popular concept that used in the probability that the total fraction will not exceed $1$and everything will be calculated under the number $0 - 1$ as zero is the least possible outcome and one is the highest outcome)
Hence, we used this concept to find the person Living probability, that is $P(\overline A ) = 1 - P(A) = 1 - \dfrac{8}{{13}} \Rightarrow \dfrac{5}{{13}}$ (from the person die probability we found the person living probability)
If we divide the probability value and multiplied with the number hundred, then we will get the percentage value for the required result.