someone is using a loaded die. the probability of rolling a \1\ is 0.5 and the rest of the outcomes have equal probabilities. find the expected value for rolling this die.

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Accepted Answer

# Explanation:
## Step1: Find the probability of each non - 1 outcome
The sum of all probabilities of a discrete probability distribution is $1$. Let $p$ be the probability of each non - 1 outcome. There are $5$ non - 1 outcomes.
We know that $P(1)+5p = 1$. Given $P(1)=0.5$, then $0.5 + 5p=1$.
Solving for $p$:
$$
\begin{align*}
5p&=1 - 0.5\
5p&=0.5\
p&=\frac{0.5}{5}=0.1
\end{align*}
$$

## Step2: Use the formula for expected value $E(X)=\sum_{i}x_ip_i$
For a die, $x_i$ are $1,2,3,4,5,6$ and $p_1 = 0.5$, $p_2=p_3=p_4=p_5=p_6 = 0.1$
$$
\begin{align*}
E(X)&=(1\times0.5)+(2\times0.1)+(3\times0.1)+(4\times0.1)+(5\times0.1)+(6\times0.1)\
&=0.5+(0.2 + 0.3+0.4+0.5+0.6)\
&=0.5 + 2\
&=2.5
\end{align*}
$$

# Answer:
$2.5$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the probability of each non - 1 outcome

Step2: Use the formula for expected value \(E(X)=\sum_{i}x_ip_i\)

Answer: