Question

if a seed is planted, it has a 50% chance of growing into a healthy plant. if 100 randomly selected seeds are planted, answer the following.
a) which is the correct wording for the random variable?
b) pick the correct symbol: n = 100
c) pick the correct symbol: p = 0.5
d) what is the probability that exactly 44 of them grow into a healthy plant?
round final answer to 4 decimal places.
e) what is the probability that less than 44 of them grow into a healthy plant?
round final answer to 4 decimal places.
f) what is the probability that more than 44 of them grow into a healthy plant?
round final answer to 4 decimal places.
g) what is the probability that exactly 54 of them grow into a healthy plant?
round final answer to 4 decimal places.
h) what is the probability that at least 54 of them grow into a healthy plant?
round final answer to 4 decimal places.
i) what is the probability that at most 54 of them grow into a healthy plant
round final answer to 4 decimal places.

Explanation:

Step1: Identify the distribution

This is a binomial distribution problem. Let \(X\) be the number of seeds that grow into healthy plants. The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success on a single - trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\). Here, \(n = 100\) and \(p=0.5\).

Step2: Answer part a

The random variable is "The number of the 100 randomly - selected seeds that grow into a healthy plant".

Step3: Answer part b

The correct symbol for the number of trials \(n = 100\).

Step4: Answer part c

The correct symbol for the probability of success on a single trial \(p = 0.5\).

Step5: Answer part d

For \(k = 44\), \(C(100,44)=\frac{100!}{44!(100 - 44)!}\), \(P(X = 44)=C(100,44)\times(0.5)^{44}\times(0.5)^{100 - 44}\)
\[

$$\begin{align*} C(100,44)&=\frac{100!}{44!×56!}\\ P(X = 44)&=\frac{100!}{44!×56!}\times(0.5)^{44}\times(0.5)^{56}\\ &\approx0.0485 \end{align*}$$

\]

Step6: Answer part e

\(P(X\lt44)=\sum_{k = 0}^{43}C(100,k)\times(0.5)^{k}\times(0.5)^{100 - k}\). Using a binomial probability calculator or software, \(P(X\lt44)\approx0.1587\)

Step7: Answer part f

\(P(X\gt44)=1 - P(X\leq44)=1-(P(X\lt44)+P(X = 44))\approx1-(0.1587 + 0.0485)=0.7928\)

Step8: Answer part g

For \(k = 54\), \(C(100,54)=\frac{100!}{54!(100 - 54)!}\), \(P(X = 54)=C(100,54)\times(0.5)^{54}\times(0.5)^{100 - 54}\)
\[

$$\begin{align*} C(100,54)&=\frac{100!}{54!×46!}\\ P(X = 54)&=\frac{100!}{54!×46!}\times(0.5)^{54}\times(0.5)^{46}\\ &\approx0.0485 \end{align*}$$

\]

Step9: Answer part h

\(P(X\geq54)=\sum_{k = 54}^{100}C(100,k)\times(0.5)^{k}\times(0.5)^{100 - k}\approx0.1587\)

Step10: Answer part i

\(P(X\leq54)=\sum_{k = 0}^{54}C(100,k)\times(0.5)^{k}\times(0.5)^{100 - k}\approx0.8413\)

Answer:

a. The number of the 100 randomly - selected seeds that grow into a healthy plant
b. \(n = 100\)
c. \(p = 0.5\)
d. \(0.0485\)
e. \(0.1587\)
f. \(0.7928\)
g. \(0.0485\)
h. \(0.1587\)
i. \(0.8413\)