# Probability of same Balls or same Numbers

I'm really getting muddled over this probability question and will be glad if I'm assisted. '' There are 5 black balls and 3 green balls in a basket. The black balls are numbered 1, 2, 3, 4 , 5 respectively and the green balls numbered 1, 2, 3 respectively. If 2 balls are to drawn random without replacement, find the probaility that these balls have either the same number or same color. So this was the equation I used. p ( same color or same number) = p ( same number) + p(same color ) - p ( their intersection). How do i move on?

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There are 3 ways that they have the same number. There are choose(5,2)=10 ways of picking two black balls. There are choose(3,2)=3 ways of picking two green balls. All of those three events are disjoint so that gives 3+10+3 = 16 ways to meet the criterion. There are a total of choose(8,2)=28 ways to pick 2 balls out of 8 without replacement. So the probability that the 2 balls drawn at random without replacement have either the same color or the same number is 16/28 = 4/7.

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There are $\binom{8}{2}$ ways to choose $2$ balls. These ways are all equally likely.

Now count the number of ways the balls can have the same number: Clearly there are $3$.

There are $\binom{5}{2}$ to choose $2$ black, and $\binom{3}{2}$ ways to choose $2$ green. So the number of "favourable" cases is $$3+\binom{5}{2}+\binom{3}{2}.$$ Divide by $\binom{8}{2}$ to find the probability.

Remark: One can also use your analysis. Note that "same colour" and "same number" are disjoint events. They cannot happen at the same time, so the probability of the intersection is $0$.

The probability of "same number" is $\frac{3}{\binom{8}{2}}$. The probability of "same colour" is $\frac{\binom{5}{2}+\binom{3}{2}}{\binom{8}{2}}$. Add.

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I prefer to solve this question by first finding neither nor condition. The total possible ways= 8*7=56

For two balls to be having neither same color nor same number

Lets select the first ball from the basket of black balls and the second one from green balls

For 1,2,and 3 of black basket we have two possibilities each from green basket For 4 and 5 of black basket, we have three possibilities each from green basket

So, a total of 12 possibilities So our answer is 56-12/56 = 44/56 = 11/14

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