Question

Given events $J\text{and}K:P\left(J\right)=0.18;P\left(K\right)=0.37;P\left(J\text{OR}K\right)=0.45$

a. Find $P\left(J\text{AND}K\right)$.

b. Find the probability of the complement of event$\left(J\text{AND}K\right)$

c. Find the probability of the complement of event$\left(J\text{OR}K\right)$

Short answer

a. The probability of $P(J\mathrm{OR}K)=0.1$

b. The probability of the complement of event $P\left((\mathrm{J}\mathrm{AND}\mathrm{K}{)}^{\text{'}}\right)=0.9$

c. The probability of the complement of event$P\left((\mathrm{J}\mathrm{OR}\mathrm{K}{)}^{\text{'}}\right)=0.55$

Step-by-step solution

Definition of Probability

The probability of an event occurring. The ratio of the total number of conceivable outcomes to the number of outcomes in an exhaustive collection of equally likely options that create a specific occurrence.

(a) We have to find the probability that occured at the same time (part a)

$$\begin{gathered} P(JANDK) \\ =P\left(K\right)+P\left(J\right)-P(JORK) \\ =0.18+0.37-0.45 \\ =0.1 \end{gathered}$$

(b) The probability of the complement of event  can be found as (part b)

$P\left((J\mathrm{AND}K{)}^{\text{'}}\right)$

$$\begin{gathered} =1-P(JANDK) \\ =1-0.1 \\ =0.9 \end{gathered}$$

(c) The probability of a complement of the event is (part c)

$P\left((\mathrm{J}\mathrm{OR}\mathrm{K}{)}^{\text{'}}\right)$

$$\begin{gathered} =1-P(JORK) \\ =1-0.45 \\ =0.55 \end{gathered}$$