Question
Question: The probability that a ship safely reaches a port is \( \dfrac{1}{3} \) . The probability that out o...
The probability that a ship safely reaches a port is 31 . The probability that out of 5 ships, at least 4 ships would arrive safely is
a) 2431
b) 24310
c) 24311
d) 24313
Solution
Hint : First we will find the probability of a ship that does not reach port safely. Then we will find probability that at least 4 arrivals are safe, for that reason we have to find p(4) and p(5) respectively. We will find 5C4(31)4(32) for p(4) and 5C5(31)5 p(5). Then we will add both the terms to get the answer.
Formula used:
If a random experiment is done k- times and the probability that an event will occur is p and the probability of that event do not occur is q, then the probability that the event will occur exactly ‘n’ times is given by the formula,
kCnpnqk−n
Where, kCn=n!×(k−n)!k!
Complete step-by-step answer :
Here we are given that the probability that a ship safely reaches a port is 31 .
So, to find the probability that it does not reach safely is basically 1− the probability that it reaches safely.
Let s denotes safe reaching and s’ denotes not reaching safe,
⇒P(s)=31
⇒P(s′)=1−31=32
Now we will find the probability that at least 4 arrives safely. We have a total 5 ships. So at least 4 arrives safely means 4 out of 5 arrives safely and 5 out of 5 arrives safely.
P(at least 4 arrive safely) = P(4) + P(5)
So we will find out the probability of both the cases separately and find out the values and then we will add these numbers.
For 4 out of 5 case =5C4(31)432
=4!×(5−4)!5!(31)432
=4!×1!5!×341×32
=5×341×32
=5×352
=3510
=3510 --(1)
For 5 out of 5 case =5C5(31)5
=5!×(5−5)!5!(31)5
=5!×0!5!(31)5
=1×(31)5
=(31)5 --(2)
Now adding (1) and (2) we get the probability of at least four ships reach safely,
=3510+351=3511=24311
So the probability that at least four ships arrive safely is 24311 .
So, the correct answer is “ 24311 (OPTION C) ”.
Note : If we are provided with probability of a certain thing and we have to find the probability of another one then we have to subtract that number from 1 to get the probability of the same. The probability of any random experiment is always a positive term and can never exceed 1.