Counting Outcomes Outcomes with Replacement Draw a card from a deck, replace it,
ID: 3171686 • Letter: C
Question
Counting Outcomes
Outcomes with Replacement
Draw a card from a deck, replace it, and shuffle the deck. There are 52 equally likely outcomes on each draw.
P(draw ace of hearts) = 1/52 is the same from draw to draw.
For each of the 52 ways in which the 1st draw occurs., there are 52 ways the 2nd draw occurs. So, there are 52 x 52 ways two draws can occur. P(acefollowed by ace) = 4/52 x 4/52
Outcomes without Replacement
Draw a card from a deck and not replace it. P(heart) = 13/52.
P(heart followed by spade) = 13/52 x 13/51
Permutations
A permutation is a sequence of outcomes where outcomes are drawn without replacement and order is important.
For a sequence of n things drawn from N things without replacement, there are N ways the 1st draw occurs.
For each way of the 1st trial, there are N-1 ways the 2nd trial can occur. For 2 draws, there are N x (N-1) possible ways. Continuing, there are N x (N-1) x (N-2) triples for 3 draws. In general, it's N!/(N - n)! , or =PERMUT( N, n)
Combinations
A combination is a group of outcomes where outcomes are drawn without replacement and order is not important.
Example: How many groups of 4 can be formed from 7 people? We know there are 840 sequences possible. But
4 things can arrange themselves 4 x 3 x 2 x 1 = 4! (read 4 factorial) ways. so, there are 840 / 4! groups. In general, the number of combinations = the number of permutations/the number of sequences of n things =N! / (n! x (N - n)!),or =COMBIN(N,n)
Sequences of 4 of 7 things 840
Groups of 4 of 7 things 35
# of poker hands 2,598,960
QUESTIONS:
1. The number of sequences (order is important) formed by drawing without replacement of 4 things from 8 things=
2. The number of sequences (order is important) formed by drawing with replacement of 4 things from 8 things=
3. The number of committees of size 4 formed from 8 people =
Explanation / Answer
1. The number of sequences (order is important) formed by drawing without replacement of 4 things from 8 things=
COMBIN(8,4) = 8! / (4! x (8 - 4)!) = 70
2. The number of sequences (order is important) formed by drawing with replacement of 4 things from 8 things=
An item can be drawn from 8 things in 8C1 = 8 ways
Hence required number of sequences are 8^4 = 4096
3. The number of committees of size 4 formed from 8 people =
COMBIN(8,4) = 8! / (4! x (8 - 4)!) = 70