Question

Two marbles are drawn one by one from a well- shuffled box containing 2 gold marbles, 6 silver marbles and 9 bronze balls. (correct to 4 decimal places)a) find the probability that they are both gold marbles if the first marble is replaced;b) find the probability that they are both silver marbles if the first marble is not replaced.

Answer 1

a) 0.0138

b) 0.1103

Explanations:

Probability is the likelihood or chance that an event will occur. Mathematically;

`Probability=(n(E))/(n(S))`

where:

n(S) is the total outcome

n(E) is the total number of events

If a well- shuffled box contains 2 gold marbles, 6 silver marbles and 9 bronze balls, the total outcome is given as:

n(S) = 2 + 6 + 9

n(S) = 17 marbles

a) If the marbles selected are two gold marbles and replaced, the probability of selecting two gold marbles will be:

`\begin{gathered} Pr(2\text{ gold marbles})=(2)/(17)*(2)/(17) \\ Pr(2\text{ gold marbles})=(4)/(289) \\ Pr(2\text{ gold marbles})\approx0.0138 \end{gathered}`

b) If two silver marbles are selected, the probability of selceting the first silver marble is given as:

`Pr(first\text{ silver marble})=(6)/(17)`

If the first silver marble is not replaced, the probability of picking the second one will be:

`Pr(second\text{ silver marble})=(6-1)/(17-1)=(5)/(16)`

The probability that they are both silver marbles if the first marble is not replaced is calculated as:

`\begin{gathered} Pr(2\text{ silver marbles})=\frac{\cancel{6}^3}{17}*\frac{5}{\cancel{16}^8} \\ Pr(2\text{ silver marbles})=(15)/(136) \\ Pr(2\text{ silver marbles})\approx0.1103 \end{gathered}`

Hence the probability that they are both silver marbles if the first marble is not replaced is 0.1103