1 of 8 0-day streak

Exact probability of repeated independent successes

Question 1 of 8

Calculator allowed

A calibration check is carried out independently for each reading

The trials are independent. The probability of a reading within the stated range on each reading is \(\frac{1}{4}\), and the probability of a reading outside the stated range is \(1-\frac{1}{4}\). The process is repeated for 7 readings

Let \(X\) be the number of successes. Find the exact probability of at least 2 successes. Give an exact probability; an equivalent fraction or terminating decimal is acceptable

(a)1 mark

Which exact term set and calculation match \(\Pr(X\ge 2)\)?

Multiply the event terms: r = 2, 3, 4, 5, 6, 7

\binom{7}{2}\left(\frac{1}{4}\right)^{2}\left(1-\frac{1}{4}\right)^{5}\times \binom{7}{3}\left(\frac{1}{4}\right)^{3}\left(1-\frac{1}{4}\right)^{4}\times \binom{7}{4}\left(\frac{1}{4}\right)^{4}\left(1-\frac{1}{4}\right)^{3}\times \binom{7}{5}\left(\frac{1}{4}\right)^{5}\left(1-\frac{1}{4}\right)^{2}\times \binom{7}{6}\left(\frac{1}{4}\right)^{6}\left(1-\frac{1}{4}\right)^{1}\times \binom{7}{7}\left(\frac{1}{4}\right)^{7}\left(1-\frac{1}{4}\right)^{0}

Use only the term for r = 2

\binom{7}{2}\left(\frac{1}{4}\right)^{2}\left(1-\frac{1}{4}\right)^{5}

Subtract from 1 the terms for r = 0, 1

1 - (\binom{7}{0}\left(\frac{1}{4}\right)^{0}\left(1-\frac{1}{4}\right)^{7} + \binom{7}{1}\left(\frac{1}{4}\right)^{1}\left(1-\frac{1}{4}\right)^{6})

Use the stated success counts, with the same exponent on the success and failure factors

1 - (\binom{7}{0}\left(\frac{1}{4}\right)^{0}\left(1-\frac{1}{4}\right)^{0} + \binom{7}{1}\left(\frac{1}{4}\right)^{1}\left(1-\frac{1}{4}\right)^{1})

Use the stated success counts, but do not count the possible positions of the successes separately

1 - (\left(\frac{1}{4}\right)^{0}\left(1-\frac{1}{4}\right)^{7} + \left(\frac{1}{4}\right)^{1}\left(1-\frac{1}{4}\right)^{6})

(b)5 marks

For the representative term with \(r=2\), complete the exact formula data

n (number of trials) r (number of successes) C(n,r) success exponent failure exponent

Calculate \(\Pr(X\ge 2)\)

Exact probability

Format: exact fraction or terminating decimal