Question 12.11E4: Planting Trees The probability that a tree planted by a land......
Planting Trees
The probability that a tree planted by a landscaping company will survive is 0.8. Determine the probability that
a) none of four trees planted will survive.
b) at least one of four trees planted will survive.
a) Success is a tree survives. Thus, p = 0.8 and q = 1 – p = 1 – 0.8 = 0.2. We want to find the probability of 0 successes in 4 trials. Thus, x = 0 and n = 4. We find the probability of 0 successes, or P(0), as follows.
P(x) = (_nC_x)p^xq^{n-x}
P(0) = (_4C_0)(0.8)^0(0.2)^{4-0}
= 1(1)(0.2)^4
= 1(1)(0.0016)
= 0.0016
Thus, the probability that none of the four trees planted will survive is 0.0016.
b) The probability that at least one tree of the four trees planted will survive can be found by subtracting from 1 the probability that none of the four trees survives. We worked problems of this type in earlier sections of the chapter, including Sections 12.6 and 12.10.
In part (a), we determined the probability that none of the four trees planted survives is 0.0016. Thus,
\text{P(at least one tree planted will survive) = }1 – P\left(\begin{matrix}\text{none of the four trees}\\ \text{planted will survive} \end{matrix} \right)
= 1 – 0.0016
= 0.9984