Question

A random sample of $415$children from England and the United States who completed a survey in a recent year was selected. Each student’s country of origin was recorded along with which superpower they would most like to have: the ability to fly, ability to freeze time, invisibility, super strength, or telepathy (ability to read minds). The data are summarized in the two-way table.

Suppose we randomly select one of these students. Define events E: England, T: telepathy, and S: superstrength.

a. Find $P\left(T\right|E).$Interpret this value in context.

b. Given that the student did not choose superstrength, what’s the probability that this child is from England? Write your answer as a probability statement using correct symbols for the events.

Short answer

Part (a) Probability for the child from England prefers Telepathy, $P\left(T\right|E)=0.22$

Part (b) Probability that the child from England did not choose superstrength,$P\left(E\right|S)\approx 0.4839$

Step-by-step solution

Part (a) Step 1. 

Data for superpowers in the two – way table:

Part (a) Step 2. Explanation

According to conditional probability,

$P\left(B\right|A)=\frac{P(A\cap B)}{P\left(A\right)}=\frac{P(AandB)}{P\left(A\right)}$

We know

E: England

T: Telepathy

Note that

The information about $415$children is provided in the table.

Thus,

The number of possible outcomes is $415$.

Also note that

In the table, $200$of the $415$children are from England.

Thus,

The number of favorable outcomes is $200$.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

$P\left(E\right)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}=\frac{200}{415}$

Now,

Note that

In the table, $44$of the $415$children are from England and prefer Telepathy. In this case, the number of favorable outcomes is $44$and number of possible outcomes is $415$.

$P(EandT)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}=\frac{44}{415}$

Apply conditional probability:

$$\begin{gathered} P\left(E\right|T)=\frac{P(EandT)}{P\left(E\right)} \\ =\frac{{\displaystyle \frac{44}{415}}}{{\displaystyle \frac{200}{415}}} \\ =\frac{44}{200} \\ =\frac{11}{50} \\ =0.2 \\ =22\% \end{gathered}$$

Therefore,

Around $22\%$of the children from England prefer Telepathy and the probability for the child from England prefers Telepathy is $0.22$.

Part (b) Step 1. Explanation

According to complement rule,

$P\left(A^c\right)=P\left(notA\right)=1-P\left(A\right)$

According to conditional probability,

$P\left(B\right|A)=\frac{P(A\cap B)}{P\left(A\right)}=\frac{P(AandB)}{P\left(A\right)}$

We know

E: England

S: Superstrength

Note that

The information about $415$children is provided in the table.

Thus,

The number of possible outcomes is $415$.

Also note that

In the table, $43$of the $415$children from both the countries prefer superstrength.

Thus,

The number of favorable outcomes is $43$.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

$P\left(S\right)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}=\frac{43}{415}$

Apply complement rule:

$P\left(S^c\right)=P\left(notS\right)=1-\frac{43}{415}=\frac{372}{415}$

Now,

Note that

In the table, $20$children from total $200$children from England prefer Superstrength.

That means

Remaining $180$children from total $200$children from England do not prefer Superstrength.

In this case, the number of favorable outcomes is $180$.

Since the total children from both the countries are $415$.

Thus,

The number of possible outcomes is $415$.

$P(Eand{S}^c)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}=\frac{180}{415}$

Apply conditional probability:

$$\begin{gathered} P\left(E\right|S^c)=\frac{P(Eand{S}^c)}{P\left(A\right)} \\ =\frac{{\displaystyle \frac{180}{415}}}{{\displaystyle \frac{372}{415}}} \\ =\frac{180}{372} \\ =\frac{15}{31} \\ \approx 0.4839 \\ =48.39\% \end{gathered}$$

Therefore,

Around $48.39\%$children not preferring Superstrength are from England and the probability for child from England did not prefer Superstrength is $0.4839$.