Question

What is the probability that a randomly chosen subject completes more than the expected number of puzzles in the $5$-minute period while listening to soothing music?

a. $0.1$

b. $0.4$

c. $0.8$

d. $1$

e. Cannot be determined

Short answer

The probability of a randomly selected subject completing more tasks than expected in a 5-minute period while listening to relaxing music is $0.40$.

Step-by-step solution

Given Information

The probability distribution is:

$X$	$1$	$2$	$3$	$4$
$P\left(x\right)$	$0.20$	$0.40$	$0.30$	$0.10$

Explanation of the correct option

While listening to smoothing music, the expected number of puzzles in 5 minutes can be calculated as follows:

$\begin{array}{rcl}E\left(x\right)& =& \sum xp\left(x\right)\\ & =& 1(0.2)+2(0.4)+3(0.3)+4(0.1)\\ & =& 0.2+0.8+0.9+0.4\\ & =& 2.3\end{array}$

Thus, the likelihood that a randomly selected subject completes more problems than expected in a 5-minute period while listening to calming music can be computed as follows:

$\begin{array}{rcl}P(X& >& 2.3)=P(X=3)+P(X=4)\\ & =& 0.30+0.10\\ & =& 0.40\end{array}$

Therefore, the correct option is " b".

Explanation for incorrect option

(a) The likelihood that a randomly selected participant completes more tasks than expected in a 5-minute period while listening to relaxing music will not be equal to $0.1.$

(c) The likelihood that a randomly selected participant completes more tasks than expected in a 5-minute period while listening to relaxing music will not be equal to $0.8$.

(d) The likelihood that a randomly selected participant completes more tasks than expected in a 5-minute period while listening to relaxing music will not be equal to $1$.

(e) The likelihood that a randomly selected participant completes more tasks than expected in a 5-minute period while listening to relaxing music Cannot be determined.