Probability of streak of events without replacement

A bag of 20 balls. 5 of each blue, green, red and yellow.

If I pick 5 balls in a row, what is the probability of picking balls of same color ?

I don't replace balls back into the bag.

I want to know how should I solve problem like this.

• Hint: the color of the first ball doesn't matter. The probability that the second one matches the first is then $\frac 4{19}$. Conditioned on the first two matching, the probability the third also matches is $\frac 3{18}$. And so on.
– lulu
Feb 8 '17 at 13:06

When you pick your first ball, the colour doesn't matter, it only sets the requirements for the next picks. Therefore we start calculating the probability after the first pick.

There are now $19$ balls left and only $4$ of the colour you need. Hence the probability of you picking a same-coloured ball is:

$$P_{\,2}=\frac{4}{19}$$

Now there are $18$ balls in the bag, but $3$ of the colour you need.

$$P_{\,3}=\frac{3}{18}=\frac{1}{9}$$

Repeating the steps from above you get:

$$P_{\,4}=\frac{2}{17};\quad P_{\,5}=\frac{1}{16}$$

The final probability is equal to the product:

$$P=\frac{4}{19}\times\frac{1}{9}\times\frac{2}{17}\times\frac{1}{16}=\frac{8}{46512}=\frac{1}{5814}$$