Tree Diagrams - A bag contains 10 counters 6 of them are white. A counter is taken at random and NOT replaced. Another counter is taken out of the bag at random.Work out the probability that only one of the two counters are white

Given that the bag contains a total of 10 counters and 6 of them are white, then 4 must be not white. Therefore, the probabilities for the first event are:

[tex]\sf P(White)=\dfrac{6}{10}\\\\\\\sf P(Not\;white)=\dfrac{4}{10}[/tex]

As the first counter is not replaced, the bag now contains 9 counters.

If the first counter was white, then the probabilities for the second event are:

[tex]\sf P(White)=\dfrac{6-1}{10-1}=\dfrac{5}{9}\\\\\\\sf P(Not\;white)=\dfrac{4-0}{10-1}=\dfrac{4}{9}[/tex]

If the first counter was not white, then the probabilities for the second event are:

[tex]\sf P(White)=\dfrac{6-0}{10-1}=\dfrac{6}{9}\\\\\\\sf P(Not\;white)=\dfrac{4-1}{10-1}=\dfrac{3}{9}[/tex]

Create a tree diagram with the given information (see attachment).

The end of each branch is labelled with its outcome, and the probability of that event is written along the branch.

To calculate the probability that only one of the two counters is white, we need to determine the probability of "White and Not white" and "Not White and White", and then add these probabilities together.

To calculate the probability of "White and Not white" we multiply the corresponding probabilities along the branches:

[tex]\sf P(White\;and\;Not\;white)=\dfrac{6}{10} \times \dfrac{4}{9}=\dfrac{24}{90}[/tex]

To calculate the probability of "Not white and White" we multiply the corresponding probabilities along the branches:

[tex]\sf P(Not\;white\;and\;White)=\dfrac{4}{10} \times \dfrac{6}{9}=\dfrac{24}{90}[/tex]

The probability that only one of the two counters is white is "White and Not white" OR "Not white and White". Therefore, we need to add these probabilities:

[tex]\sf \dfrac{24}{90}+\dfrac{24}{90}=\dfrac{48}{90}=\dfrac{8}{15}=53.3\%\;(nearest\;tenth)[/tex]

So, the probability that only one of the two counters is white is 8/15 which is approximately 53.3% (rounded to the nearest tenth).