1 of 8 0-day streak

Exact probability of repeated independent successes

Question 1 of 8

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A laboratory procedure is repeated independently for each sample

The trials are independent. The probability of the specified result on each run is \(\frac{2}{5}\), and the probability of another result is \(1-\frac{2}{5}\). The process is repeated for 10 runs

Let \(X\) be the number of successes. Find the exact probability of at least 4 successes. Give an exact probability; an equivalent fraction or terminating decimal is acceptable

(a)1 mark

Which exact term set and calculation match \(\Pr(X\ge 4)\)?

Subtract from 1 the terms for r = 0, 1, 2, 3

1 - (\binom{10}{0}\left(\frac{2}{5}\right)^{0}\left(1-\frac{2}{5}\right)^{10} + \binom{10}{1}\left(\frac{2}{5}\right)^{1}\left(1-\frac{2}{5}\right)^{9} + \binom{10}{2}\left(\frac{2}{5}\right)^{2}\left(1-\frac{2}{5}\right)^{8} + \binom{10}{3}\left(\frac{2}{5}\right)^{3}\left(1-\frac{2}{5}\right)^{7})

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

1 - (\left(\frac{2}{5}\right)^{0}\left(1-\frac{2}{5}\right)^{10} + \left(\frac{2}{5}\right)^{1}\left(1-\frac{2}{5}\right)^{9} + \left(\frac{2}{5}\right)^{2}\left(1-\frac{2}{5}\right)^{8} + \left(\frac{2}{5}\right)^{3}\left(1-\frac{2}{5}\right)^{7})

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

1 - (\binom{10}{0}\left(\frac{2}{5}\right)^{0}\left(1-\frac{2}{5}\right)^{0} + \binom{10}{1}\left(\frac{2}{5}\right)^{1}\left(1-\frac{2}{5}\right)^{1} + \binom{10}{2}\left(\frac{2}{5}\right)^{2}\left(1-\frac{2}{5}\right)^{2} + \binom{10}{3}\left(\frac{2}{5}\right)^{3}\left(1-\frac{2}{5}\right)^{3})

Multiply the event terms: r = 4, 5, 6, 7, 8, 9, 10

\binom{10}{4}\left(\frac{2}{5}\right)^{4}\left(1-\frac{2}{5}\right)^{6}\times \binom{10}{5}\left(\frac{2}{5}\right)^{5}\left(1-\frac{2}{5}\right)^{5}\times \binom{10}{6}\left(\frac{2}{5}\right)^{6}\left(1-\frac{2}{5}\right)^{4}\times \binom{10}{7}\left(\frac{2}{5}\right)^{7}\left(1-\frac{2}{5}\right)^{3}\times \binom{10}{8}\left(\frac{2}{5}\right)^{8}\left(1-\frac{2}{5}\right)^{2}\times \binom{10}{9}\left(\frac{2}{5}\right)^{9}\left(1-\frac{2}{5}\right)^{1}\times \binom{10}{10}\left(\frac{2}{5}\right)^{10}\left(1-\frac{2}{5}\right)^{0}

Use only the term for r = 4

\binom{10}{4}\left(\frac{2}{5}\right)^{4}\left(1-\frac{2}{5}\right)^{6}

(b)5 marks

For the representative term with \(r=4\), complete the exact formula data

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

Calculate \(\Pr(X\ge 4)\)

Exact probability

Format: exact terminating decimal