Question

If a seed is planted, it has a 60% chance of growing into a healthy plant. If 100 seeds are planted,

(a) find the probability that exactly 55 will grow into a healthy plant:  

(b) find the probability that at least 54 will grow into a healthy plant:  

(c) find the probability that fewer than 67 will grow into a healthy plant:  

(d) find the probability that at most 63 will grow into a healthy plant:  

Round all answers to four decimals.

How would I show my work for these?

Answer 1

Solution:
n= 100
p^ = 60%
=0.60
We can calculate the mean and standard deviation of random variable x as,
μ=μx = np^ = 100× 0.6 = 60
σx =✓[ np^ (1-p^) ]
=✓[ 100×0.6 ×(1-0.4) ]
=4.8990
P(X=55)= P{[ (X - μ) /σ]= [(55 - 60)/ 4.8990]}
=P(Z= -1.02)

P(X=55)=0.1539
b)P(X>=54)= 1- P(X<54)
=1-P{[ (X - μ) /σ]< [(54 - 60)/ 4.8990]}   
=1- P(Z= -1.22)
=1- 0.1112
P(X>=54)= 0.8888
c) P(X<67)= P{[ (X - μ) /σ]< [(67 - 60)/ 4.8990]}   
= P(Z< 1.43)
P(X<67) =0.9236
d) P(X<=63)= P{[ (X - μ) /σ]<=[(63 - 60)/ 4.8990]}   
= P(Z<= 0.61)
P(X<=63)=0.7291