Single probability inequality

6 The residents of Mahjing were asked to classify their local bus service:

• \(25 \%\) of residents classified their service as good.
• \(60 \%\) of residents classified their service as satisfactory.
• \(15 \%\) of residents classified their service as poor.
  1. A random sample of 110 residents of Mahjing is chosen.

Use a suitable approximation to find the probability that fewer than 22 residents classified their bus service as good.

For a random sample of 10 residents of Mahjing, find the probability that fewer than 8 classified their bus service as good or satisfactory.

Three residents of Mahjing are selected at random. Find the probability that one resident classified the bus service as good, one as satisfactory and one as poor.

2 In a large college, \(32 \%\) of the students have blue eyes. A random sample of 80 students is chosen. Use an approximation to find the probability that fewer than 20 of these students have blue eyes.

3 A factory produces a certain type of electrical component. It is known that \(15 \%\) of the components produced are faulty. A random sample of 200 components is chosen. Use an approximation to find the probability that more than 40 of these components are faulty.

1 It is known that, on average, 2 people in 5 in a certain country are overweight. A random sample of 400 people is chosen. Using a suitable approximation, find the probability that fewer than 165 people in the sample are overweight.

2 Assume that, for a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 350 randomly chosen people, at least 47 will have their next birthday on a Monday.

3 On a production line making cameras, the probability of a randomly chosen camera being substandard is 0.072 . A random sample of 300 cameras is checked. Find the probability that there are fewer than 18 cameras which are substandard.

2 When visiting the dentist the probability of waiting less than 5 minutes is 0.16 , and the probability of waiting less than 10 minutes is 0.88 .

1. Find the probability of waiting between 5 and 10 minutes. A random sample of 180 people who visit the dentist is chosen.
2. Use a suitable approximation to find the probability that more than 115 of these people wait between 5 and 10 minutes.

2 The probability that George goes swimming on any day is \(\frac { 1 } { 3 }\). Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.

3 It is found that \(10 \%\) of the population enjoy watching Historical Drama on television. Use an appropriate approximation to find the probability that, out of 160 people chosen randomly, more than 17 people enjoy watching Historical Drama on television.

2 On a production line making toys, the probability of any toy being faulty is 0.08 . A random sample of 200 toys is checked. Use a suitable approximation to find the probability that there are at least 15 faulty toys.

1 When a butternut squash seed is sown the probability that it will germinate is 0.86 , independently of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the probability that more than 210 germinate.

3 A fair dice is thrown 90 times. Use an appropriate approximation to find the probability that the number 1 is obtained 14 or more times.

1 The random variable \(F\) has the distribution \(B ( 50,0.7 )\). Use a suitable approximation to find \(\mathbf { P } \boldsymbol { ( } \mathbf { F > } \mathbf { 4 0 } \boldsymbol { ) }\). [5]

3 Sixty people each make two throws with a fair six-sided die.

1. State the probability of one particular person obtaining two sixes.
2. Using a suitable approximation, calculate the probability that at least four of the sixty obtain two sixes.

3 A factory produces 9000 music DVDs each day. A random sample of 100 such DVDs is obtained.

1. Explain how to obtain this sample using random numbers.
2. Given that \(24 \%\) of the DVDs produced by the factory are classical, use a suitable approximation to find the probability that, in the sample of 100 DVDs, fewer than 20 are classical.

4. A random variable \(X\) has the distribution \(\mathrm { B } ( 80,0.375 )\).

1. Write down the mean and variance of \(X\).
2. Use the Normal approximation to the binomial distribution to estimate \(\mathrm { P } ( X > 40 )\).

2 The random variable \(W\) has the distribution \(B \left( 40 , \frac { 2 } { 7 } \right)\). Use an appropriate approximation to find \(\mathrm { P } ( W > 13 )\).

A disease occurs in 3\% of a population.

1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution. [2]
2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people. [3]
3. Find the mean and variance of the number of people with the disease in a random sample of 100 people. [2]

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

1. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination. [3]

In a large town, 35% of the inhabitants have access to television channel \(C\). A random sample of 60 inhabitants is obtained. Use a suitable approximation to find the probability that 18 or fewer inhabitants in the sample have access to channel \(C\). [6]

The random variable \(F\) has the distribution B\((40, 0.65)\). Use a suitable approximation to find P\((F \leq 30)\), justifying your approximation. [7]

In a town, 54% of the residents are female and 46% are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female. [4 marks]