The random variable X represents the number of successes in 10 independent Bernoulli trials - HSC - SSCE Mathematics Extension 1 - Question 6 - 2021 - Paper 1
Question 6
The random variable X represents the number of successes in 10 independent Bernoulli trials. The probability of success is p = 0.9 in each trial.
Let r = P(X ≥ 1).
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Worked Solution & Example Answer:The random variable X represents the number of successes in 10 independent Bernoulli trials - HSC - SSCE Mathematics Extension 1 - Question 6 - 2021 - Paper 1
Step 1
Calculate P(X ≥ 1)
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Answer
To find r, we calculate the probability that at least one success occurs in 10 trials. This can be calculated as:
r=P(X≥1)=1−P(X=0)
The probability of getting zero successes (X = 0) in 10 trials where p = 0.9 can be calculated using the formula for the binomial distribution:
P(X=0)=(1−p)n=(1−0.9)10=0.110=10−10
Thus, we find:
r=1−10−10≈0.9999999999
This shows that r is very close to 1.
Step 2
Determine the value of r
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Answer
Given that r is approximately 0.9999999999, it is evident that:
r > 0.9
Since r is very close to 1, we can confidently say:
The correct choice that describes the value of r is A: r > 0.9.
