Question

An urn contains 100 marbles: 20 white, 30 red, 50 green. Calculate the probability of selecting White, Red and Green marbles respectively. What is the probability of pulling a white, green, white and red marbles consecutively?

Answer 1

`950/156849` or approximately `0.6%`

Explanation:

Assuming the marbles are not replaced in the urn:

• The probability of the first marble being white is `20/100`

• The probability of the next marble being green is then `50/99`

• The probability of the next marble being white is `19/98`

• The probability of the next marble being red is `30/97`

So the probability of the sequence white, green, white, red is:

`20/100 * 50/99 * 19/98 * 30/97`

`=10/color(red)(cancel(color(black)(50)))*color(red)(cancel(color(black)(50)))/99*19/98*30/97`

`=(10*19*30)/(99*98*97)`

`=(color(red)(cancel(color(black)(3)))*1900)/(color(red)(cancel(color(black)(3)))*33*98*97)`

`=1900/(33*98*97)`

`=(color(red)(cancel(color(black)(2)))*950)/(33*color(red)(cancel(color(black)(2)))*49*97)`

`=950/(33*49*97)`

`=950/156849 ~~ 0.006`

That is approximately `0.6%`

Answer 2

In support of Georg's solution

Explanation:

For probability questions of this type, if you are ever in doubt, draw a probability tree

`color(red)("Assumption: this is selection without replacement")`

From the diagram observe that the initial selection of
`White ->20/100->20%`

`Red" "-> 30/100->30%`

`Green->50/100->50%`

From the probability tree the overall sequenced sampling probability of white: green: white: red is:

`20/100xx50/99xx19/98xx30/97`

For what follows refer to George's solution