An experiment consists of tossing a coin 14 times.
(a) How many different outcomes are possible?
(b) How many different outcomes have exactly 9 heads?
(c) How many different outcomes have at least 2 heads?
(d) How many different outcomes have at most 10 heads?
To answer these questions, we need to understand the concept of combinations and permutations.
Permutations represent the number of different arrangements of a set of objects, where the order of the objects matters. Combinations, on the other hand, represent the number of different ways you can choose a subset of objects from a larger set, where the order doesn't matter.
(a) How many different outcomes are possible?
In this case, when tossing a coin 14 times, each toss can result in two different outcomes: heads (H) or tails (T).
Since each toss is independent, we can use the multiplication principle to find the total number of outcomes.
Number of different outcomes = number of outcomes for each toss^number of tosses
= 2^14
= 16,384
Therefore, there are 16,384 different possible outcomes.
(b) How many different outcomes have exactly 9 heads?
To find the number of outcomes with exactly 9 heads, we need to calculate the number of combinations of 9 heads out of 14 tosses. We can use the combination formula:
Number of combinations = nCr = n! / (r!(n-r)!)
Where n is the total number of items (14 tosses) and r is the number of items we are choosing (9 heads).
Number of combinations = 14C9 = 14! / (9!(14-9)!)
= 2002
Therefore, there are 2002 different outcomes with exactly 9 heads.
(c) How many different outcomes have at least 2 heads?
To find the number of outcomes with at least 2 heads, we need to calculate the total number of outcomes and subtract the outcomes with 0 or 1 head.
Total number of outcomes = 2^14 = 16,384
Outcomes with 0 head = 1 (all tails)
Outcomes with 1 head = 14 (one head and any of the remaining 13 tosses being tails)
Number of outcomes with at least 2 heads = Total number of outcomes - Outcomes with 0 head - Outcomes with 1 head
= 16,384 - 1 - 14
= 16,369
Therefore, there are 16,369 different outcomes with at least 2 heads.
(d) How many different outcomes have at most 10 heads?
To find the number of outcomes with at most 10 heads, we need to calculate the sum of outcomes with 0, 1, 2, ..., 10 heads.
Number of outcomes with at most 10 heads = Number of outcomes with 0 head + Number of outcomes with 1 head + ... + Number of outcomes with 10 heads
Using the formula from part (c):
Number of outcomes with at most 10 heads = 1 + 14 + 14C2 + 14C3 + ... + 14C10
Calculate the sum using the combination formula for each term, and then add them together.
Finally, find the sum:
Number of outcomes with at most 10 heads = 1 + 14 + 91 + 364 + 1001 + 2002 + 3003 + 3432 + 3003 + 2002 + 1001
= 13,965
Therefore, there are 13,965 different outcomes with at most 10 heads.