Hypergeometric Distribution

3 Andy has 4 red socks and 8 black socks in his drawer. He takes 2 socks at random from his drawer.

1. Find the probability that the socks taken are of different colours.
   The random variable \(X\) is the number of red socks taken.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

7 A bag contains \(2 m\) yellow and \(m\) green counters. Three counters are chosen at random, without replacement. The probability that exactly two of the three counters are yellow is \(\frac { 28 } { 55 }\). Determine the value of \(m\).

3 Adila has a pack of 50 cards.

1. Each of the 50 cards is numbered with a different integer from 1 to 50 . Adila selects 5 cards at random without replacement.
   (a) Find the probability that exactly 3 of the 5 cards have numbers which are 10 or less.
   (b) Adila arranges the 5 cards in a line in a random order. Find the probability that the 5 cards are arranged in numerically increasing order. 10 of the 50 cards are blue and the rest are green.
2. Adila randomly selects three sets of 10 cards each, without replacement. The sets are labelled \(A , B\) and \(C\). Given that \(A\) contains 3 blue cards and 7 green cards, find the probability that \(B\) contains exactly 2 blue cards and \(C\) contains exactly 3 blue cards.

1 A bottle of sweets contains 13 red sweets, 13 blue sweets, 13 green sweets and 13 yellow sweets. 7 sweets are selected at random. Find the probability that exactly 3 of them are red.

3 A pet shop has 6 rabbits and 3 hamsters. 5 of these pets are chosen at random. The random variable \(X\) represents the number of hamsters chosen.

1. Show that the probability that exactly 2 hamsters are chosen is \(\frac { 10 } { 21 }\).
2. Draw up the probability distribution table for \(X\).

7 At a garden centre there is a box containing 50 hyacinth bulbs. Of these, 30 will produce a blue flower and the remaining 20 will produce a red flower. Unfortunately they have become mixed together so that it is not known which of the bulbs will produce a blue flower and which will produce a red flower. Karen buys 3 of these bulbs.

1. Find the probability that all 3 of these bulbs will produce blue flowers.
2. Find the probability that Karen will have at least one flower of each colour from her 3 bulbs.

4 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.

1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
2. Draw up the probability distribution table for \(X\).
3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).

1. Anju has a bag that contains 5 socks of which 2 are blue.

Anju randomly selects socks from the bag, one sock at a time. She does not replace any socks but continues to select socks at random until she has both blue socks. The discrete random variable \(S\) represents the total number of socks that Anju has selected.

1. Write down the value of \(\mathrm { P } ( S = 1 )\)
2. Find \(\mathrm { P } ( S > 2 )\)
3. Find \(\mathrm { P } ( S = 3 )\)
4. Given that the second sock selected is blue, find the probability that Anju selects exactly 3 socks.
5. Find \(\mathrm { P } ( S = 5 )\)

3 A pack of 40 cards consists of 10 cards in each of four colours: red, yellow, blue and green. The pack is dealt at random into four "hands", each of 10 cards. The hands are labelled North, South, East and West.

1. Find the probability that West has exactly 3 red cards.
2. Find the probability that West has exactly 3 red cards, given that East and West have between them a total of exactly 5 red cards.
3. South has 5 red cards and 5 blue cards. These cards are placed in a row in a random order. Find the probability that the colour of each card is different from the colour of the preceding card.

2 A bag contains 5 red balls and 3 blue balls. Sadie takes 3 balls at random from the bag, without replacement. The random variable \(X\) represents the number of red balls that she takes.

1. Show that the probability that Sadie takes exactly 1 red ball is \(\frac { 15 } { 56 }\).
2. Draw up the probability distribution table for \(X\).
3. Given that \(\mathrm { E } ( X ) = \frac { 15 } { 8 }\), find \(\operatorname { Var } ( X )\).

1. Julia selects 3 letters at random, one at a time without replacement, from the word

\section*{VARIANCE} The discrete random variable \(X\) represents the number of times she selects a letter A.

1. Find the complete probability distribution of \(X\). Yuki selects 10 letters at random, one at a time with replacement, from the word \section*{DEVIATION}
2. Find the probability that he selects the letter E at least 4 times.