Answer to Question #346997 in Statistics and Probability for Nzuzi

Question

Question 1.9 [2, 2, 3]

Chantelle has decided to sell baked biscuits to assist in the payment of her university fees. After

baking for hours and packing packets to sell, she finds that she has 9 biscuits left over. Of these 9

biscuits, 4 are chocolate biscuits, 3 are raisin and 2 are peanut butter. She thinks to herself that she

is going to use these 9 biscuits to assist her with understanding probability. She treats each biscuit

as being slightly different, however order of her selection is not important.

Suppose Chantelle selects 3 biscuits at random from the 9, help her answer the following questions:

a) Calculate the probability that of the 3 biscuits randomly selected, 1 is chocolate, 1 is raisin

and 1 is peanut butter.

b) Calculate the probability that only chocolate biscuits are selected

c) Calculate the probability that at least

Answer 1

a)

$$P(3\ different)=\dfrac{\dbinom{4}{1}\dbinom{3}{1}\dbinom{2}{1}}{\dbinom{9}{3}}$$

$$=\dfrac{4(3)(2)}{84}=\dfrac{2}{7}$$

b)

$$P(3\ chocolate)=\dfrac{\dbinom{4}{3}\dbinom{3}{0}\dbinom{2}{0}}{\dbinom{9}{3}}$$

$$=\dfrac{4(1)(1)}{84}=\dfrac{1}{21}$$

c)

$$P(at\ least\ 1\ chocolate)=1-P(No\ chocolate)$$

$$=1-\dfrac{\dbinom{4}{0}\dbinom{3+2}{3}}{\dbinom{9}{3}}$$

$$=1-\dfrac{1(10)}{84}=\dfrac{37}{42}$$