Step 1a

Finding the value of 'k'

Explanation: Sum of probabilities = 1,

X is a random variable.

\$\sum_{i=0}^6 P(X) = 1\$

P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6) = 1

\$0 + k + 2k + 4k + k^2 + 2k^2 + 3k^2 = 1\$

\$7k + 6k^2 = 1\$

\$6k^2 + 7k - 1 = 0\$

\$6k^2 + 6k + k - 1 = 0\$

\$6k(k+1)-1(k+1) = 0\$

\$(6k-1)(k+1) = 0\$

\$k = 1/6 , k = -1\$

k is probability and it cannot be negative.

\$k = 1/6 \$

Step 1b

Determine P(x<3)

Explanation: P(x<3) = P(X=0) + P(X=1) + P(X=2)

⇒0 + k + 2k

⇒ 3k

⇒ \$3 × 1/6\$

\$P(x<3) = 1/2\$

\$P(x<3) = 0.5\$
i.e, < 1

Step 1c

Determine P(0<x<3)

Explanation: P(0<x<3) = P(X=1) + P(X=2)

⇒ k + 2k

⇒ 3k

⇒ \$3 × 1/6\$

\$P(0<x<3) = 1/2\$

\$P(0<x<3) = 0.5\$
i.e, < 1

Step 1d

Determine P(x≥5)

Explanation: P(x≥5) = P(X=5) + P(X=6)

⇒\$2k^2 + 3k^2\$

⇒\$5k^2\$

⇒\$5(1/6)^2\$

⇒\$5 × 1/36\$

\$P(x≥5) = 5/36\$

\$P(x≥5) = 0.13\$
i.e,< 1