Question

Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of 1 (low) to 5 (high). Let X be the pain score for a randomly selected patient. The following table gives part of the probability distribution for X.

Value	$1$	$2$	$3$	$4$	$5$
Probability	$0.1$	$0.2$	$0.3$	$0.3$	?

(a) Find $P(X=5)$

(b) If two patients who received artificial knees are chosen at random, what’s the probability that both of them report pain scores of $1$or $2$? Show your work.

(c) Compute the mean and standard deviation of $X$. Show your work.

Short answer

(a)$P(X=5)=0.1$

(b)$P\left(1\text{or}2\right)=0.09$

(c) Mean=$3.1$

Standard deviation=$1.14$

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that, Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of $1$(low) to $5$(high). Let $X$be the pain score for a randomly selected patient.

We need to find $P(X=5)$

Part (a) Step 2: Explanation

Given:

the probability distribution is

Value	$1$	$2$	$3$	$4$	$5$
Probability	localid="1649746499147" $0.1$	localid="1649746505085" $0.2$	localid="1649746511170" $0.3$	localid="1649746516856" $0.3$	??

The formulas to compute the mean and standard deviation are:

localid="1649746524407" $\sigma =\sqrt{\sum x^2\times P\left(x\right)-{\left(\sum x\times P\left(x\right)\right)}^2}$

Let localid="1649746530834" $\mathrm{x}$be the missing value.

The missing value can be calculated as:

localid="1649746539988" $0.1+0.2+0.3+0.3+x=1$

localid="1649746546770" $x+0.9=1$

localid="1649746553075" $x=0.1$

localid="1649746559408" $P(X=5)$can be calculated as:

localid="1649746565386" $P(X=5)=x$

localid="1649746571448" $=0.1$

Part (b) Step 1: Given information

Given in the question that, Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of $1$(low) to $5$(high). Let$X$be the pain score for a randomly selected patient.

We need to find the probability that both of the patients scores either $1$or $2$

Part (b) Step 2: Explanation

The probability that both of the patients scores either$1$or $2$is computed as:

$P\left(1\text{or}2\right)=P(X=1)+P(X=2)$

$=0.1+0.2$

$=0.3$

The probability that both of the patients scores either$1$or $2$is computed as:

$P(1$or $2$$)=P(1$or $2$$)\times P(1$or $2)$$)$

$=0.3\times 0.3$

$=0.09$

Part (c) Step 1: Given information

Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of $1$(low) to $5$(high). Let $X$be the pain score for a randomly selected patient.

We need to compute the mean and standard deviation of X .

Part (c) Step 2: Explanation

The mean can be calculated as:

$\mathrm{Mean}=\sum x\times P\left(x\right)$

$=1\left(0.1\right)+2\left(0.2\right)+3\left(0.3\right)+4\left(0.3\right)+5\left(0.1\right)$

localid="1649746701130" $=3.1$

The standard deviation is calculated as follows:

localid="1649746705865" $\sigma =\sqrt{\sum x^2\times P\left(x\right)-{\left(\sum x\times P\left(x\right)\right)}^2}$

localid="1649746709064" $=\sqrt{1^2\times 0.1+2^2\times 0.2+\dots .+5^2\times 0.2-(3.1{)}^2}$

localid="1649746712012" $=1.1358$