Please could someone explain this probability question!

• Oct 21st 2011, 04:10 AM
Ben91
Question :
Suppose a train driver has a 2% chance of crash during every journey. Can we say that he would be mathematically certain of crash after 45 journeys?

I am trying to apply the combinations formula in this question as a tree diagram would be too time consuming, I am now thinking i am going down the COMPLETELY wrong path!(Doh) Could anyone please take a few minutes to give me some guidance, thank you!!(Happy)
• Oct 21st 2011, 04:58 AM
emakarov
Re: Help with probability question please!
If A and B are two events, let $\displaystyle A\land B$ denote the event when both A and B occur, and $\displaystyle A\lor B$ denote the event when either A or B (or both) occur. Then $\displaystyle P(A\lor B)=P(A)+P(B)-P(A\land B)$. So, in general the probability of $\displaystyle A\lor B$ is smaller than the sum of probabilities.

On the other hand, if A and B are independent, then $\displaystyle P(A\land B)=P(A)P(B)$. So, if crashes during different journeys are independent, then the probability that there will be no crash during 45 journeys is $\displaystyle (1-0.02)^{45}$.
• Oct 21st 2011, 11:27 AM
Soroban
Re: Help with probability question please!
Hello, Ben91!

Quote:

Suppose a train driver has a 2% chance of crash during every journey.

Can we say that he would be mathematically certain of crash after 45 journeys?

No!

The key phrase is "every journey".

For each journey: .$\displaystyle \begin{Bmatrix}P(\text{crash}) &=& 0.02 \\ P(\text{no crash}) &=& 0.98 \end{Bmatrix}$

We have:
. . $\displaystyle P(\text{No crashes in 342 journeys}) \;=\;(0.98)^{342} \;\approx\;0.001$

Therefore:
. . $\displaystyle P(\text{at least one crash in 342 journeys}) \;=\;1 - 0.001 \;=\;0.999 \;=\;99.9\%$

We can never be 100% certain of a crash.

• Oct 21st 2011, 01:01 PM
Ben91
Re: Help with probability question please!
As usual, I was completely over thinking the question! Big thank you to both of you for explaining the question and allowing me to understand :)