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 Probability r
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Answer
To find the probability that there is at least one success (r = P(X ≥ 1)), we can use the complement rule:
P(X ≥ 1) = 1 - P(X = 0).
First, we need to calculate P(X = 0), the probability of getting zero successes in 10 trials. This can be computed using the formula for a binomial distribution:
P(X=0)=(1−p)n=(0.1)10=0.0000000001.
Now substituting this back into the complement:
P(X≥1)=1−P(X=0)=1−0.0000000001≈1.
Step 2
Determine Value of r
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Answer
From the calculation, we find that r is extremely close to 1.
Thus, it is valid to conclude:
Since r is close to 1, option A (r > 0.9) is the most accurate description of the value of r.
