Questions: A weighted number cube has faces labeled 1 through 6. Let X be the number shown on the cube when it is rolled. Here is the probability distribution of X. Value x of X: 1, 2, 3, 4, 5, 6 P(X=x): 0.18, 0.28, 0.16, 0.15, 0.12, 0.11 For parts (a) and (b) below, find the probability that the number described is shown on a random roll. (a) Greater than or equal to 5: (b) Lower than 3:

A weighted number cube has faces labeled 1 through 6. Let X be the number shown on the cube when it is rolled. Here is the probability distribution of X.

Value x of X: 1, 2, 3, 4, 5, 6
P(X=x): 0.18, 0.28, 0.16, 0.15, 0.12, 0.11

For parts (a) and (b) below, find the probability that the number described is shown on a random roll.
(a) Greater than or equal to 5:
(b) Lower than 3:

Transcript text: A weighted number cube has faces labeled 1 through 6 . Let $X$ be the number shown on the cube when it is rolled. Here is the probability distribution of $X$. \begin{tabular}{|c|c|c|c|c|c|c|} \hline Value $\boldsymbol{x}$ of $\boldsymbol{X}$ & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline $\boldsymbol{P}(\mathbf{X}=\boldsymbol{x})$ & 0.18 & 0.28 & 0.16 & 0.15 & 0.12 & 0.11 \\ \hline \end{tabular} For parts (a) and (b) below, find the probability that the number described is shown on a random roll. (a) Greater than or equal to 5: $\square$ (b) Lower than 3: $\square$

Solution

Solution Steps

Step 1: Identify the probabilities for values greater than or equal to 5

From the probability distribution table:

$ P(X = 5) = 0.12 $
$ P(X = 6) = 0.11 $

Step 2: Calculate the probability for part (a)

The probability that $ X $ is greater than or equal to 5 is the sum of the probabilities for $ X = 5 $ and $ X = 6 $: \[ P(X \geq 5) = P(X = 5) + P(X = 6) = 0.12 + 0.11 = 0.23 \]

Step 3: Identify the probabilities for values lower than 3

From the probability distribution table:

$ P(X = 1) = 0.18 $
$ P(X = 2) = 0.28 $

Step 4: Calculate the probability for part (b)

The probability that $ X $ is lower than 3 is the sum of the probabilities for $ X = 1 $ and $ X = 2 $: \[ P(X < 3) = P(X = 1) + P(X = 2) = 0.18 + 0.28 = 0.46 \]

Final Answer

(a) $ \boxed{0.23} $
(b) $ \boxed{0.46} $

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