Problem Solving

probability of raining on any day in a week is 50 % .what is the prob that it rains exactly 3 days in a span of 5 days?

There are �C₃ = 10 different 3-day combinations from a group of 5 days.

Probability of raining first 3 days out of 5 = (0.5)^3 * (1 - 0.5)^2 = 0.03125. We know that there are 10 such outcomes.

Therefore, actual probability = 10 * 0.03125 = 0.3125.

I could be way off the mark on this; can someone confirm? Experts?

It can rain in 5C3 ways in 5 days = 10 ways.
the probability that it rains on any three days = 1/2 * 1/2 * 1/2 = 1/8 ways
thw probability that it does not rain on any 2 days = 1/2 * 1/2 = 1/4
Total probability of raining on exactly any three days = 10 * 1/8 * 1/4 = 5 /16 = .3125

@ edge: Please don't convert fractions into decimals. It will make calculations harder for you.

if we calculate the compliment probability, i.e. it doesn't rain exactly two days in a span of five days, then we have 5C2=10 combination sets,
we get P(2 days doesn't rain)=((1/2)^5 )*10=5/16 which means P(3 days rain)

winniethepooh wrote:It can rain in 5C3 ways in 5 days = 10 ways.
the probability that it rains on any three days = 1/2 * 1/2 * 1/2 = 1/8 ways
thw probability that it does not rain on any 2 days = 1/2 * 1/2 = 1/4
Total probability of raining on exactly any three days = 10 * 1/8 * 1/4 = 5 /16 = .3125

@ edge: Please don't convert fractions into decimals. It will make calculations harder for you.
You probably forgot to multiply your result by 10.

ruplun wrote:probability of raining on any day in a week is 50 % .what is the prob that it rains exactly 3 days in a span of 5 days?

P(exactly n times) = P(one way) * total possible ways.

Let R = rain and N = no rain.

P(one way):
One way to get exactly 3 days of rain is to have rain on the first 3 days but not on the last 2 days.
P(RRRNN) = 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32.

Total possible ways:
Any arrangement of the letters RRRNN will yield exactly 3 days of rain.
Thus, to account for all the ways to get exactly 3 days of rain, the result above needs to be multiplied by the number of ways to arrange RRRNN.
Number of ways to arrange RRRNN = 5!/(3!2!) = 10.

Multiplying the results above, we get:
P(exactly 3 days of R) = 10 * 1/32 = 10/32 = 5/16.

For an even trickier rain problem, check here:

https://www.beatthegmat.com/rain-check-t79099.html

winniethepooh wrote:@ edge: Please don't convert fractions into decimals. It will make calculations harder for you. You probably forgot to multiply your result by 10.

Huh? I multiplied by 10...

The explanation and solution provided by GMATGuruNY is very good ... it makes the reasoning and picture far clearer....

edge wrote:

winniethepooh wrote:@ edge: Please don't convert fractions into decimals. It will make calculations harder for you. You probably forgot to multiply your result by 10.

Huh? I multiplied by 10...

Oops, my bad!! I missed that!
Edited above.