Answer: 0.756
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Step-by-step explanation:
We have two possible events A and B
A = Stan gets 2 or more questions correct
B = Stan gets none correct, or Stan gets exactly one correct
Finding P(A) is a lot of work doing so directly without using P(B). Why? Because we need to find P(2), P(3), P(4), ... all the way up to P(10). Then we add up all those results. The notation P(x) means the probability of getting x correct.
If we find P(B) first, then we can use the fact that
P(A) + P(B) = 1
P(A) = 1 - P(B)
which is a more efficient route.
Note how A and B are complementary events
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Let's find P(0) which is the probability of getting x = 0 correct
n = 10
x = 0
n C x = 10 C 0 = (10!)/(0!*(10-0)!) = 1
P(x) = (n C x)*(0.25)^x*(1-0.25)^(n-x)
P(x) = (n C x)*(0.25)^x*(0.75)^(n-x)
P(x) = (10 C x)*(0.25)^x*(0.75)^(10-x)
P(0) = (10 C 0)*(0.25)^0*(0.75)^(10-0)
P(0) = (1)*(0.25)^0*(0.75)^10
P(0) = 0.05631351470948
Repeat for x = 1 as well
n = 10
x = 1
n C x = 10 C 1 = (10!)/(1!*(10-1)!) = 10
P(x) = (n C x)*(0.25)^x*(1-0.25)^(n-x)
P(x) = (n C x)*(0.25)^x*(0.75)^(n-x)
P(x) = (10 C x)*(0.25)^x*(0.75)^(10-x)
P(1) = (10 C 1)*(0.25)^1*(0.75)^(10-1)
P(1) = (10)*(0.25)^1*(0.75)^9
P(1) = 0.18771171569824
So we see that,
P(B) = getting 0 correct or getting 1 correct
P(B) = P(0) + P(1)
P(B) = 0.05631351470948+0.18771171569824
P(B) = 0.24402523040772
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Now we can find P(A)
P(A) = probability of getting at least two correct
P(A) = 1 - P(B)
P(A) = 1 - 0.24402523040772
P(A) = 0.75597476959229
P(A) = 0.756
The probability of getting at least two correct is roughly 0.756