The probability that a train arrives late at a particular station is 2

Question

The probability that a train arrives late at a particular station is 25%. There are four trains scheduled to arrive at the station today. What is the probability that at least three of them will arrive later than the scheduled time?

A) \(\frac{1}{256}\)

B) \(\frac{5}{256}\)

C) \(\frac{13}{256}\)

D) \(\frac{20}{256}\)

E) \(\frac{255}{256}\)

A)  \frac{1}{256}

B)  \frac{5}{256}

C)  \frac{13}{256}

D)  \frac{20}{256}

E)  \frac{255}{256}

chetan2u · Answer

Hi,
 Probability that a train is late =  \frac{25}{100} =\frac{1}{4} , so prob of train reaching in time =1-1/4=  \frac{3}{4}   
atleast THREE means
1) all four..
 (\frac{1}{4})^4=\frac{1}{256} 
2) Only three..
 \frac{1}{4}*\frac{1}{4}*\frac{1}{4}*\frac{3}{4}*4=\frac{12}{256} 
multiplication by 4 is as the train coming in time can be any of the 4

total  \frac{12+1}{256}=\frac{13}{256}

c

SVaidyaraman · Answer

1. There are 4 ways of 3 trains coming late. Probability of 3 trains coming late and one train not coming late is (1/4*1/4*1/4*3/4)*4=12/256
2. Probability of 4 trains coming late is 1/4*1/4*1/4*1/4=1/256
3. Final probability is 13/256

ScottTargetTestPrep · Answer

We are given that the probability that a train arrives late is 1/4.
 
We need to determine the probability that at least 3 trains arrive late. That is, we need to determine the probability that exactly 3 trains arrive late or that all 4 trains arrive late.
 
Let’s start with 3 late trains and 1 on-time train (L = late and T = on time):
 
P(L-L-L-T) = 1/4 x 1/4 x 1/4 x 3/4 = 3/256
 
Since we can organize the 3 Ls and 1 T in 4!/3! = 4 ways, the total probability is actually 3/256 x 4 = 12/256.
 
Next we can determine the probability that all 4 trains arrive late:
 
P(L-L-L-L) = 1/4 x 1/4 x 1/4 x 1/4 = 1/256
 
Thus, the probability of at least 3 trains arriving late is 12/256 + 1/256 = 13/256.
 
Answer: C

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The probability that a train arrives late at a particular station is 2

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