Normal approximation to binomial

5 The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams.

1. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams.
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2. The shop also sells red apples. \(60 \%\) of the red apples sold by the shop weigh more than 80 grams. 160 red apples are chosen at random from the shop. Use a suitable approximation to find the probability that fewer than 105 of the chosen red apples weigh more than 80 grams.

6 The diameters of apples in an orchard have a normal distribution with mean 5.7 cm and standard deviation 0.8 cm . Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.

1. Find the probability that an apple selected at random can be used as a toffee apple.
2. 250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.

7 In tests on a new type of light bulb it was found that the time they lasted followed a normal distribution with standard deviation 40.6 hours. 10\% lasted longer than 5130 hours.

1. Find the mean lifetime, giving your answer to the nearest hour.
2. Find the probability that a light bulb fails to last for 5000 hours.
3. A hospital buys 600 of these light bulbs. Using a suitable approximation, find the probability that fewer than 65 light bulbs will last longer than 5130 hours.

6 Ana meets her friends once every day. For each day the probability that she is early is 0.05 and the probability that she is late is 0.75 . Otherwise she is on time.

1. Find the probability that she is on time on fewer than 20 of the next 96 days.
2. If she is early there is a probability of 0.7 that she will eat a banana. If she is late she does not eat a banana. If she is on time there is a probability of 0.4 that she will eat a banana. Given that for one particular meeting with friends she does not eat a banana, find the probability that she is on time.