Binomial distribution Combnations

Binomial distribution KUS objectives BAT use factorial notation to find the number of arrangements of some objects Starter: Discuss how many ways are there: Of choosing a Captain and deputy captain from 11 players Of getting 3 heads and two tails when you flip a coin 5 times Of winning the main UK lottery? How sure are you of your answers?

This pattern is known as Notes2 Consider the number of combinations from tossing coins: 1 coin Outcome No. of combinations 1 pick This pattern is known as Pascal’s Triangle 2 coins Outcome No. of combinations 2 picks 3 coins Outcome No. of combinations 3 picks 4 coins Outcome No. of combinations 4 picks 5 coins Outcome No. of combinations 5 picks It gives the number of combinations that give r matches from n picks between 2 options (in this case heads or tails) Eg how many ways can you get 3 tails out of 4 tosses? 4 ways

Eg how many ways can you get 3 heads out of 5 tosses? Notes 3 This calculation is known as Pascal realised that the number of combinations could be directly calculated from n and r: or Number of combinations and has a calculator key Eg how many ways can you get 3 heads out of 5 tosses? Possible combinations: HHHTT HHTTH HTTHH TTHHH HHTHT HTHTH THTHH HTHHT THHTH Eg how many ways can you get 2 heads out of 5 tosses? THHHT This is reflected by the fact that Pascal’s triangle is symmetrical Why are these the same? If r = n – r, the calculation gives the same result ie substitute r = n – r in the rule for nCr Number of combinations

Total number of possible arrangements Where does the rule come from? For example, why does ? ie how many ways can you obtain 3 heads and 2 tails with 5 coins? Call the 3 heads H1, H2 and H3 and the two tails T1 and T2 Any of the 5 could be the first you toss Any of the remaining 4 could be the second Etc until last position Hence 5 x 4 x 3 x 2 x 1 = 5! ways to arrange the 5 coins But we don’t care which Head came 1st, 2nd or 3rd and which tail came 1st or 2nd There are 2! ways to order the tails and 3! ways to order the heads, giving 2! x 3! ‘duplicates’ of HHHTT Consider the ways to obtain HHHTT: The same is true for any other ways of obtaining 3 heads and 2 tails (eg HHTHT) It follows that if = the number of distinct outcomes: Duplicates Desired, distinct outcomes Total number of possible arrangements

Firstly: n different objects can be arranged in n! ways Notes5 summary Firstly: n different objects can be arranged in n! ways Secondly: n objects with r of one type and (n - r) of another can be arranged in 𝒏𝑪𝒓= 𝒏 𝒓 = 𝒏! 𝒓! 𝒏−𝒓 ! ways

RRBB BBRR RBBR BRRB RBRB BRBR WB 1 Find all the possible arrangements of Three objects where one is red, one is blue and one is green Four objects where two are red, and two are blue First object can be chosen in 3 ways; 2nd object can be chosen in two ways and the third object can be chosen in one way 3!=3×2×1=6 𝑤𝑎𝑦𝑠 RBG RGB BRG BGR GRB GBR b) If we treat the objects as four different things 𝑅 1 , 𝑅 2 , 𝐵 1 , 𝐵 2 then there are 4!=24 ways. However, arrangements with 𝑅 1 𝑅 2 are identical to those with 𝑅 2 𝑅 1 … and similarly with 𝐵 1 𝐵 2 and 𝐵 1 𝐵 2 So there are 4! 2!2! =6 𝑤𝑎𝑦𝑠 RRBB BBRR RBBR BRRB RBRB BRBR

A child has 4 yellow and 6 black identically shaped wooden bricks. WB 2 A child has 4 yellow and 6 black identically shaped wooden bricks. The bricks are assembled into a vertical tower one brick deep. Find the number of different patterns of the bricks which can be formed Ten bricks with 4Y and 6B can arranged in 𝟏𝟎! 𝟒!𝟔! = 210 ways

A bag has three £1 coins, two 50p coins and four 10p coins. WB 3 A bag has three £1 coins, two 50p coins and four 10p coins. Three coins are randomly selected. Find the probability that the value of the coins is a) £2.50 b) £1.60 c) 30p £2.50 must be made of two x £1 and one 50p → 𝟑! 𝟐!𝟏! = 3 ways The probability is given by: P(£1) x P(£1/ £1 drawn) x P(50p/ two £1 coins drawn) x 3 ways = 3 9 × 2 8 × 2 7 ×3= 1 14 b) £1.60 must come from £1, 50p and 10p → 3! ways The probability is given by 3 9 × 2 8 × 4 7 ×3!= 2 7 c) 30p must come from 10p, 10p and 10p → 𝟑! 𝟑! = 1 way The probability is given by 4 9 × 3 8 × 2 7 ×1= 1 21

Practice Ex 1A

One thing to improve is – KUS objectives BAT use factorial notation to find the number of arrangements of some objects self-assess One thing learned is – One thing to improve is –

END