A student randomly guesses on 10 true or false questions. Use the binomial model
to determine the probability that the student gets 5 out of the 10 questions right.
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P (x) = [ ]
n!
x!(n−x)!
p
xq
n−x
In this case, we can use the binomial probability formula to calculate the probability that the student gets 5 out of 10 questions right:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
- P(x) is the probability of getting x successes
- n is the total number of trials (10 questions in this case)
- x is the number of successes we want (5 questions right)
- p is the probability of success on a single trial (0.5, since it's a true or false question and the student is randomly guessing)
- q is the probability of failure on a single trial (1-p, which is 0.5 in this case)
Plugging in the values, we have:
P(5) = (10C5) * (0.5^5) * (0.5^(10-5))
= (10!) / (5!(10-5)!) * (0.5^5) * (0.5^5)
= (362880) / (5!(120)(0.5^10)
= (362880) / ((5!)(120)(0.0009765625))
= (362880) / (120 * 0.0009765625)
= (362880) / (0.1171875)
= 3096576
Therefore, the probability that the student gets 5 out of 10 questions right is 3096576.
Probability of exactly 5 successes P(X = 5)0.24609375
Apologies for the previous incorrect calculation. The correct probability that the student gets exactly 5 out of 10 questions right is P(X = 5) = 0.24609375.
To calculate this, we use the binomial probability formula:
P(X = 5) = (10C5) * (0.5^5) * (0.5^(10-5))
P(X = 5) = (10!)/ (5!(10-5)!) * (0.5^5) * (0.5^5)
P(X = 5) = (10!)/ (5! * 5!) * (0.5^5) * (0.5^5)
P(X = 5) = (10 * 9 * 8 * 7 * 6)/ (5 * 4 * 3 * 2 * 1) * (0.5^5) * (0.5^5)
P(X = 5) = 252 * (0.5^10)
P(X = 5) = 252 * 0.0009765625
P(X = 5) = 0.24609375
Therefore, the probability that the student gets exactly 5 out of 10 questions right is 0.24609375.