# What is the probability that the first $5$ balls are blue?

Four red, $$8$$ blue, and $$5$$ green balls are randomly arranged in a line. (a) What is the probability that the first $$5$$ balls are blue?

Here is my solution,

Since there are $$17$$ balls we can randomly arranged in a line, Number of ways to arrange ($$m$$) $$\dfrac{17!}{8!5!4!}$$

Since first five balls are blue number of ways($$n$$) $$\dfrac{12!}{4!3!5!}$$

required probability is $$\dfrac{n}{m}$$

• What is the result of $\frac{n}{m}$? – Snoop Apr 25 at 20:46
• Yes your working is correct. – Math Lover Apr 25 at 20:50
• @MathLover slader.com/discussion/question/… – puka Apr 25 at 20:51
• Yes it is the same thing. You could have considered them all different. Your working will give the same result. – Math Lover Apr 25 at 20:56
• @Got it thank you very much – puka Apr 25 at 20:57

Your answer is quite correct. $$\cfrac{~\cfrac{12!}{3!5!4!}~}{\cfrac{17!}{8!5!4!}}=\dfrac{8!/3!}{17!/12!}$$

This is indeed the probability for obtaining $$\mathit 5$$ from $$\mathit 8$$ blue balls when selecting any $$\mathit 5$$ from all $$\mathit 17$$ balls to place in the first five positions (I.E. that blue is the colour of all balls in the first five positions).
\begin{align}\binom {8}5\div\binom{17}5 &= \dfrac{~~8\cdot~~7\cdot~~6\cdot~~5\cdot~~4}{17\cdot 16\cdot 15\cdot14\cdot 13}\\[1ex]&=\dfrac{2}{17\cdot 13}\\[1ex]&=\dfrac{2}{221}\end{align}

This is a hypergeometric distribution with parameters 8, 5, 5. The probability that X=5 can be found via (8 choose 5) * ( 9 choose 0) / (17 choose 5).

I don’t understand your answer, most probably due to my lack of knowledge in probabilities.

However, i do understand that much:

Arranging the balls in a line is equivalent with attempting to draw balls from a bag, while discarding the ball just drawn.

1. The odds of blue at first attempt are 8 in 17
2. At second attempt are 7 in 16
3. At third attempt are 6 in 15
4. At fourth attempt are 5 in 14
5. At fifth attempt are 4 in 13

The odds of making 5 consecutive attempts resulting in blue ball drawn are the result of the product of the previous individual attempts. This answer was already presented by someone else so I concur with it. I will not provide the formula, but I will provide the final answer since this is what you asked for: $$\frac{2}{221}$$

• Quibble: The term "odds" in conjunction with the figures used is incorrect. GOK when this bad habit will be eradicated, if ever ! – true blue anil Apr 26 at 6:01

Direct calculation: $$\frac{8\times 7\times 6\times 5\times 4}{17\times 16\times 15\times 14\times 13}=0.009049773755656$$

Fraction $$=\frac{2}{221}$$.