Question

The probability that a randomly chosen person has a dog is 0.34. The probability that a randomly chosen person will have a dog and be walking in the park is 0.03. The probability that a randomly chosen person is walking in the park or has a dog is 0.40. What is the probability that a randomly chosen person is not walking in the park?
Round to the nearest hundredths

Answer 1

Taking into definition of probability, the probability that a randomly chosen person is not walking in the park is 0.91.

Definition of Probabitity

Probability is the possibility that a phenomenon or an event will happen, given certain circumstances. It is expressed as a percentage.

Union of events

The union of events, AUB, is the event formed by all the elements of A and B. That is, the event AUB is verified when one of the two, A or B, or both occurs. AUB is read as "A or B".

The probability of the union of two compatible events is calculated as the sum of their probabilities subtracting the probability of their intersection:

P(A∪B)= P(A) + P(B) -P(A∩B)

Complementary event

A complementary event is made up of the inverse of the results of another event. That is, That is, given an event A, a complementary event is verified as long as the event A is not verified.

The probability of occurrence of the complementary event A' will be:

P(A´)= 1- P(A)

Events and probability in this case

In first place, let's define the following events:

• D: a person has a dog.
• W: a person is walking in the park.

Then you know:

• P(D)= 0.34
• P(D and W)= P(D∩W)= 0.03 [The intersection of events, A ∩ B, is the event formed by all the elements that are, at the same time, from A and B. That is, the event A ∩ B is verified when A and B occur simultaneously.]
• P(D or W)= P(D∪W)= 0.40

In this case, considering the definition of union of eventes, the probability that a randomly chosen person is walking in the park is calculated as:

P(D∪W)= P(D) + P(W) -P(D∩W)

0.40= 0.34 + P(W) -0.03

Solving:

0.40= 0.31 + P(W)

0.40 - 0.31= P(W)

0.09= P(W)

Then, the probability that a randomly chosen person is walking in the park is 0.09.

Considering the definition of the complementary event and its probability, the probability that a randomly chosen person is NOT walking in the park is calculated as:

P [W']= 1- P(W)

Replacing and solving:

P [W']= 1 - 0.09

P [W']= 0.91

Finally, the probability that a randomly chosen person is not walking in the park is 0.91.

Learn more about probability:

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