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The probability that a 50-year-old man will be alive at 60 is 0.83 and the probability that a 45-year-old woman will be alive at 55 is 0.87. Then
A. The probability that both will be alive is 0.7221
B. At least one of them will be alive is 0.9779
C. At least one of them will be alive is 0.8230
D. The probability that both will be alive is 0.6320

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Last updated date: 20th Jun 2024
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Answer
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Hint: In this question, the trick of making something happen is to calculate the probability of the event not happening and subtracting that from 1 to get the probability of the event occurring. Also note that in this question the solutions are being asked after 10 years so do not confuse with ages, keep the timeframe in mind.

Formulas used:
Probability of an event happening is the probability of an event not happening subtracted from unity.
$P\left( {\bar a} \right) = 1 - P\left( a \right)$

Complete step by step solution:
In this question, first we need to find the probability of men and women both being alive after 10 years.
As such, the probability of a man being alive is given in question $P\left( M \right) = 0.83$.
Probability of women being alive is also given which is $P\left( W \right) = 0.87$
So, probability of both being alive will be
$\
  P\left( B \right) = P\left( M \right) \cdot P\left( W \right) \\
   = 0.83 \times 0.87 \\
   = 0.7221 \\
\ $
Now to find the probability of at least one of them being alive, we have to find the probability of both of them dying and then subtract it from 1 to get the answer. Let that probability be $P\left( O \right)$
Probability of man dying is
$\
  P\left( {\bar M} \right) = 1 - P\left( M \right) \\
   = 1 - 0.83 \\
   = 0.17 \\
\ $
Probability of woman dying is
$\
  P\left( {\bar W} \right) = 1 - P\left( W \right) \\
   = 1 - 0.87 \\
   = 0.13 \\
\ $
So, probability of both dying is
$\
  P\left( {\bar O} \right) = P\left( {\bar M} \right) \cdot P\left( {\bar W} \right) \\
   = 0.17 \times 0.13 \\
   = 0.0221 \\
\ $
As such, probability of at least one being alive is
$\
  P\left( O \right) = 1 - P\left( {\bar O} \right) \\
   = 1 - 0.0221 \\
   = 0.9779 \\
\ $

Hence, Option B is the correct answer.

Note:
When dealing with probability, the probability of any event occurring or not occurring will contain every possible outcome and as such is 1 and so we were able to compute an event occurring based on if the event is not occurring. In fact it is sometimes easy to find the possibility of an event not occurring so the formula above comes in very handy.