Probability Question

[bubbagump]

An unbiased coin is tossed five times. Find the probability that the coin lands heads exactly once.

AND

A world record was set in a given year for the longest run on an un-gaffed (fair) roulette wheel. The number 34 appeared 4 times in a row. What is the probability of the occurrence of this event? (Assume that there are 38 equally likely outcomes consisting of the number 1-36, 0, and 00. If you enter your answer in scientific notation, round the decimal value to two decimal places. Use equivalent rounding if you do not enter your answer in scientific notation.)

Any ideas?? Thank you!

 

[galactus]

An unbiased coin is tossed five times. Find the probability that the coin lands heads exactly once.

The coin lands 4 tails and one head in 5 flips. But, the head could occur on any of the 5 flips. So, we have a binomial:

\(\displaystyle 5\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)^{4}=5\left(\frac{1}{2}\right)^{5}\)

A world record was set in a given year for the longest run on an un-gaffed (fair) roulette wheel. The number 34 appeared 4 times in a row. What is the probability of the occurrence of this event? (Assume that there are 38 equally likely outcomes consisting of the number 1-36, 0, and 00. If you enter your answer in scientific notation, round the decimal value to two decimal places. Use equivalent rounding if you do not enter your answer in scientific notation.)

Think about it. If there are 38 outcomes, what is the probability of one particular outcome?. Wouldn't it be 1/38?.

What if this happened 4 times in a row?. Each outcome is independent of the other, so you would multiply them. See?.

What do you think it is?. Do not over think it.
This is a rare occurence. One lucky so-and-so. Though, not as lucky/rare as a Megamillion ticket.

 

[bubbagump]

I understand how you got the coin answer. That makes sense! Thanks...

However, I don't understand what you're saying for the second one...
(1/38) (1/38) (1/38) (1/38) and multiply them together?!
I'm confused...if that's not what its supposed to be...

 

[galactus]

Yes, that's it.

The probability of it occuring once is 1/38

The probability of it occuring twice in a row is (1/38)(1/38)

and so on.

Thus, 4 times in a row .......\(\displaystyle \left(\frac{1}{38}\right)^{4}=\frac{1}{2085136}\)