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. 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: Calculate $P(X = 44)$

$C(100,44)=\frac{100!}{44!(100 - 44)!}=\frac{100!}{44!56!}$
$P(X = 44)=C(100,44)\times(0.5)^{44}\times(0.5)^{56}=\frac{100!}{44!56!}\times(0.5)^{100}\approx0.0485$

Step3: Calculate $P(X\lt44)$

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

Step4: Calculate $P(X\gt44)$

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

Step5: Calculate $P(X = 54)$

$C(100,54)=\frac{100!}{54!(100 - 54)!}=\frac{100!}{54!46!}$
$P(X = 54)=C(100,54)\times(0.5)^{54}\times(0.5)^{46}=\frac{100!}{54!46!}\times(0.5)^{100}\approx0.0485$

Step6: Calculate $P(X\geq54)$

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

Step7: Calculate $P(X\leq54)$

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

Answer:

d) $0.0485$
e) $0.1587$
f) $0.7928$
g) $0.0485$
h) $0.1587$
i) $0.8413$