First slide

Introduction to probability

Question

Three randomly chosen non-negative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is

Moderate

A

$\frac{1}{2}$
B

$\frac{36}{55}$
C

$\frac{6}{11}$
D

$\frac{5}{11}$

Solution

x + y + z = 10
Total number of non-negative solutions $=^{10 + 3 - 1} C_{3 - 1}$
$=^{12} C_{2} = 66$
Now let z = 2n
$∴ x + y = 10 - 2 n; n \geq 0$
Number of non-negative solutions of above equation is
11 + 9 + 7 + 5 + 3 + 1 = 36.
$∴ Required probability = \frac{36}{66} = \frac{6}{11}$

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