General probability threshold

3 In Restaurant Bijoux 13\% of customers rated the food as 'poor', 22\% of customers rated the food as 'satisfactory' and \(65 \%\) rated it as 'good'. A random sample of 12 customers who went for a meal at Restaurant Bijoux was taken.

1. Find the probability that more than 2 and fewer than 12 of them rated the food as 'good'. On a separate occasion, a random sample of \(n\) customers who went for a meal at the restaurant was taken.
2. Find the smallest value of \(n\) for which the probability that at least 1 person will rate the food as 'poor' is greater than 0.95.

4 When people visit a certain large shop, on average \(34 \%\) of them do not buy anything, \(53 \%\) spend less than \(
) 50\( and \)13 \%\( spend at least \)\\( 50\).

1. 15 people visiting the shop are chosen at random. Calculate the probability that at least 14 of them buy something.
2. \(n\) people visiting the shop are chosen at random. The probability that none of them spends at least \(
   ) 50\( is less than 0.04 . Find the smallest possible value of \)n$.

6 The results of a survey by a large supermarket show that \(35 \%\) of its customers shop online.

1. Six customers are chosen at random. Find the probability that more than three of them shop online.
2. For a random sample of \(n\) customers, the probability that at least one of them shops online is greater than 0.95 . Find the least possible value of \(n\).
3. For a random sample of 100 customers, use a suitable approximating distribution to find the probability that more than 39 shop online.

3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and \(15 \%\) as large.

1. Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.
2. Yue-chen picks \(n\) mangoes at random. The probability that none of these \(n\) mangoes is small is at least 0.1 . Find the largest possible value of \(n\).

2 Annan has designed a new logo for a sportswear company. A survey of a large number of customers found that \(42 \%\) of customers rated the logo as good.

1. A random sample of 10 customers is chosen. Find the probability that fewer than 8 of them rate the logo as good.
2. On another occasion, a random sample of \(n\) customers of the company is chosen. Find the smallest value of \(n\) for which the probability that at least one person rates the logo as good is greater than 0.995 .

13 In this question you must show detailed reasoning. The probability that Paul's train to work is late on any day is 0.15 , independently of other days.

1. The number of days on which Paul's train to work is late during a 450-day period is denoted by the random variable \(Y\). Find a value of \(a\) such that \(\mathrm { P } ( Y > a ) \approx \frac { 1 } { 6 }\). In the expansion of \(( 0.15 + 0.85 ) ^ { 50 }\), the terms involving \(0.15 ^ { r }\) and \(0.15 ^ { r + 1 }\) are denoted by \(T _ { r }\) and \(T _ { r + 1 }\) respectively.
2. Show that \(\frac { T _ { r } } { T _ { r + 1 } } = \frac { 17 ( r + 1 ) } { 3 ( 50 - r ) }\).
3. The number of days on which Paul's train to work is late during a 50-day period is modelled by the random variable \(X\).
   (a) Find the values of \(r\) for which \(\mathrm { P } ( X = r ) \leqslant \mathrm { P } ( X = r + 1 )\).
   (b) Hence find the most likely number of days on which the train will be late during a 50-day period.