Chapter 5.3, Problem 61E

(a)

To determine

Probability for the child from England prefers Telepathy.

Answer to Problem 61E

Probability for the child from England prefers Telepathy,

  $P\left(\text{T}|\text{E}\right)=0.22$

Explanation of Solution

Given information:

Data for superpowers in the two − way table:

  

According to conditional probability,

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

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.

  $P\left(\text{E}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{200}{415}$

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.

  $P\left(\text{E}\text{and}\text{T}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{44}{415}$

Apply conditional probability:

  $P\left(\text{E}|\text{T}\right)=\frac{P\left(\text{E}\text{and}\text{T}\right)}{P\left(\text{E}\right)}=\frac{\frac{44}{415}}{\frac{200}{415}}=\frac{44}{200}=\frac{11}{50}=0.22=22\%$

Therefore,

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

(b)

To determine

Probability for the child from England did not choose superstrength.

Answer to Problem 61E

Probability that the child from England did not choose superstrength,

  $P\left(\text{E}|{\text{S}}^{\text{c}}\right)\approx 0.4839$

Explanation of Solution

Given information:

Data for superpowers in the two − way table:

  

According to complement rule,

  $P\left({\text{A}}^{\text{c}}\right)=P\left(\text{not}\text{A}\right)=1-P\left(\text{A}\right)$

According to conditional probability,

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

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.

  $P\left(\text{S}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{43}{415}$

Apply complement rule:

  $P\left({\text{S}}^{\text{c}}\right)=P\left(\text{not}\text{S}\right)=1-\frac{43}{415}=\frac{372}{415}$

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.

  $P\left(\text{E}\text{and}{\text{S}}^{\text{c}}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{180}{415}$

Apply conditional probability:

  $P\left(\text{E}|{\text{S}}^{\text{c}}\right)=\frac{P\left(\text{E}\text{and}{\text{S}}^{\text{c}}\right)}{P\left(\text{A}\right)}=\frac{\frac{180}{415}}{\frac{372}{415}}=\frac{180}{372}=\frac{15}{31}\approx 0.4839=48.39\%$

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.