Central Station: prob. at least 6 trains are on time....

[Guest]

65 % of all scheduled trains passing through Central Station arrive on time. If 8 trains go through the train every hour, find:

i. The probability that at least 6 trains are on time
ii. The probability that at least 6 trains are on time for 9 of the next 10 hours.

I don't want answers
just how to do it...thanks!

 

[galactus]

Re: Central Station

65 % of all scheduled trains passing through Central Station arrive on time. If 8 trains go through the train every hour, find:

i. The probability that at least 6 trains are on time

\(\displaystyle \L\\\sum_{k=6}^{8}\begin{pmatrix}8\\k\end{pmatrix}p^{k}q^{8-k}\)

p=0.65 and q=0.35


ii. The probability that at least 6 trains are on time for 9 of the next 10 hours.

I don't want answers
just how to do it...thanks!

 

[soroban]

Re: Central Station

Hello, americo74!

Let me baby-talk through Galactus' explanation . . .

65 % of all scheduled trains passing through Central Station arrive on time.
If 8 trains go through the train every hour, find:

a) The probability that at least 6 trains are on time during the next hour

b) The probability that at least 6 trains are on time for 9 of the next 10 hours.

(a) We want the probability that: \(\displaystyle \text{6 are on time }\) or \(\displaystyle \text{ 7 are on time }\) or \(\displaystyle \text{ 8 are on time.}\)

Find the three probabilities and add them.

\(\displaystyle \;\;P(\text{6 on time}) \:=\:\begin{pmatrix}8\\6\end{pmatrix}(0.65)^6(0.35)^2\)

\(\displaystyle \;\;P(\text{7 on time}) \:=\:\begin{pmatrix}8\\7\end{pmatrix}(0.65)^7(0.35)\)

\(\displaystyle \;\;P(\text{8 on time}) \:=\:\begin{pmatrix}8\\8\end{pmatrix}(0.65)^8\)

Their sum is some decimal \(\displaystyle p\) . . . I'll let you crank it out.

Therefore: \(\displaystyle \,P(\text{at least 6 on time}) \:=\\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

(b) During any hour: \(\displaystyle \,P(\text{6 or more on time}) \:=\\)

\(\displaystyle \;\;\;\,\)During any hour: \(\displaystyle \,P(\text{5 or less on time}) \:=\:1\,-\,p \:=\:q\)

For 9 of the next 10 hours, the probability is: \(\displaystyle \L\,\begin{pmatrix}10\\9\end{pmatrix}p^9\,q\)

 

[Guest]

thank you to both of you

u are stars!