Question

If a, b and c are three numbers (not necessarily different) chosen and with replacement from the set {1, 2, 3, 4, 5}, the probability that (ab + c) is even, is

• A. $\frac{35}{125}$
• B. $\frac{59}{125}$
• C. $\frac{64}{125}$
• D. $\frac{75}{125}$

Answer 1

The correct option is B $\frac{59}{125}$

The probability of $ab$ being odd is ${\left(\frac{3}{5}\right)}^2=\frac{9}{25}$

The probability of $ab$ being even is $1-\frac{9}{25}=\frac{16}{25}$

The probability of $c$ being odd is $\frac{3}{5}$ and being even is $\frac{2}{5}$

$ab+c$ is even if $ab$ and $c$ are both odd with probability $\frac{9}{25}\times \frac{3}{5}=\frac{27}{125}$

or $ab$ and $c$ are both odd with probability $\frac{16}{25}\times \frac{2}{5}=\frac{32}{125}$

Thus the total probability is $\frac{59}{125}$