Sampling without replacement

5 Jasmine has one \(\\) 5\( coin, two \)\\( 2\) coins and two \(\\) 1\( coins. She selects two of these coins at random. The random variable \)X$ is the total value, in dollars, of these two coins.

1. Show that \(\mathrm { P } ( X = 7 ) = 0.2\).
2. Draw up the probability distribution table for \(X\).
3. Find the value of \(\operatorname { Var } ( X )\).

2 A box contains 10 pens of which 3 are new. A random sample of two pens is taken.

1. Show that the probability of getting exactly one new pen in the sample is \(\frac { 7 } { 15 }\).
2. Construct a probability distribution table for the number of new pens in the sample.
3. Calculate the expected number of new pens in the sample.

4 A pet shop has 9 rabbits for sale, 6 of which are white. A random sample of two rabbits is chosen without replacement.

1. Show that the probability that exactly one of the two rabbits in the sample is white is \(\frac { 1 } { 2 }\).
2. Construct the probability distribution table for the number of white rabbits in the sample.
3. Find the expected value of the number of white rabbits in the sample.

4 A box contains 2 green sweets and 5 blue sweets. Two sweets are taken at random from the box, without replacement. The random variable \(X\) is the number of green sweets taken. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 A sixth-form class consists of 7 girls and 5 boys. Three students from the class are chosen at random. The number of boys chosen is denoted by the random variable \(X\). Show that

1. \(\quad \mathrm { P } ( X = 0 ) = \frac { 7 } { 44 }\),
2. \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 22 }\). The complete probability distribution of \(X\) is shown in the following table.

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)	\(\frac { 7 } { 44 }\)	\(\frac { 21 } { 44 }\)	\(\frac { 7 } { 22 }\)	\(\frac { 1 } { 22 }\)

3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

1 In her purse, Katharine has two \(\pounds 5\) notes, two \(\pounds 10\) notes and one \(\pounds 20\) note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(\pounds X\).

1. (A) Show that \(\mathrm { P } ( X = 10 ) = 0.1\).
   (B) Show that \(\mathrm { P } ( X = 30 ) = 0.2\). The table shows the probability distribution of \(X\).

   \(r\)	10	15	20	25	30
   \(\mathrm { P } ( X = r )\)	0.1	0.4	0.1	0.2	0.2

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.

1. Show that the probability that exactly one of the marbles is yellow is \(\frac { 5 } { 14 }\).
   The random variable \(X\) is the number of yellow marbles selected.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.

1. Draw up the probability distribution of \(X\). [3]
2. Show that \(\mathrm{E}(X) = \frac{8}{7}\) and calculate \(\mathrm{Var}(X)\). [3]
3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac{3}{5}\) or the geese with probability \(\frac{2}{5}\). If the dog chases the ducks there is a probability of \(\frac{1}{10}\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac{1}{4}\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese. [4]