First slide

Introduction to probability

Question

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

Moderate

A

$\frac{35}{125}$
B

$\frac{59}{125}$
C

$\frac{64}{125}$
D

$\frac{75}{125}$

Solution

P(number chosen is odd) = 3/5
P(number chosen is even)=2/5
$\begin{array}{l} E: (a b + c) is even note that even E can occurs in two cases \\ E_{1} : all the three numbers a, b and ' c' are odd P (E_{1}) = {(\frac{3}{5})}^{3} = \frac{27}{125} \end{array}$
$E_{2} :'c' is even and at least one of a or b is even P (E_{2}) = \frac{2}{5} (1 - \frac{9}{25}) = \frac{2}{5} \cdot \frac{16}{25} = \frac{32}{125}$
$\begin{array}{l} P (E) = P (E_{1} or E_{2}) \\ = P (E_{1}) + P (E_{2}) \\ = \frac{59}{125} \end{array}$

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