Assume that p(a) = 0.4 and p(b) = 0.7. making no further assumptions on a and b, show that p (ab) satisfies 0.1 ≤ p (ab) ≤ 0.4.

The required prove is shown below:

Step-by-step explanation:

Consider the provided information

P(A) = 0.4 and P(B) = 0.7

We know that:

The maximum value of is less than or equal to 1.

......(1)

Also

Now substitute the respective values.

.....(2)

Now from equation 1 and 2 we get.

Hence, proved

(a) {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …}

(b) $59 398

step-by-step explanation:

(a) the annual sequence of profits

aₙ = a₁rⁿ⁻¹

the model for the sequence of profits is {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …} .

(b) profit after 15 yr

  a₁ = $ 30 000

  r = 1.05

a₁₅ = 30 000 × 1.05¹⁴

a₁₅ = 30 000 × 1.979 93

a₁₅ = $59 398

the profit in year 15 will be $59 398.