Probability between two values

2 Anil is a candidate in an election. He received \(40 \%\) of the votes. A random sample of 120 voters is chosen. Use an approximation to find the probability that, of the 120 voters, between 36 and 54 inclusive voted for Anil.

6 A box contains 4 pears and 7 oranges. Three fruits are taken out at random and eaten. Find the probability that

1. 2 pears and 1 orange are eaten, in any order,
2. the third fruit eaten is an orange,
3. the first fruit eaten was a pear, given that the third fruit eaten is an orange. There are 121 similar boxes in a warehouse. One fruit is taken at random from each box.
4. Using a suitable approximation, find the probability that fewer than 39 are pears.

2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.

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.

The discrete random variable \(X\) is distributed B(\(n\), \(p\)).

1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution. [1]
2. Give a reason to support your value. [1]
3. Given that \(n = 200\) and \(p = 0.48\), find P(\(90 \leq X < 105\)). [7]

A fair dice is thrown 1000 times and the number, \(X\), of throws on which the score is 6 is noted.

1. 1. State the distribution of \(X\). [1]
   2. Explain why a normal distribution would be an appropriate approximation to the distribution of \(X\). [1]
2. Use a normal distribution to find two positive integer values, \(a\) and \(b\), such that \(\text{P}(a < X < b) \approx 0.4\). [5]
3. For your two values of \(a\) and \(b\), use the distribution of part (a)(i) to find the value of \(\text{P}(a < X < b)\), correct to 3 significant figures. [2]