Question

If we flip a coin 20 times, we get a sequence of heads and tails.

(a) How many sequences of heads and tails are possible?
(b) How many sequences of heads and tails have exactly 5 heads?
(c) How many different sequences have at most 2 heads?
(d) How many sequences have at least three heads?

Answer 1

In a sequence of 20 coin flips, there are 1,048,576 possible sequences. There are 15,504 sequences with exactly 5 heads. There are 211 sequences with at most 2 heads. There are 1,048,365 sequences with at least three heads.

Step-by-step explanation:

(a) To calculate the number of possible sequences of heads and tails, we use the fundamental counting principle. Since each coin toss has 2 possible outcomes (heads or tails), the total number of sequences for flipping 20 coins is 2^20 = 1,048,576 possible sequences.

(b) To determine the number of sequences with exactly 5 heads, we use the combination formula nCk. For 5 heads, there are 20 choose 5 = 15504 possible sequences.

(c) To find the number of different sequences with at most 2 heads, we can count the sequences with 0, 1, or 2 heads. For 0 heads, there is 1 sequence. For 1 head, there are 20 sequences. For 2 heads, there are 20 choose 2 = 190 possible sequences. So, the total number of sequences with at most 2 heads is 1 + 20 + 190 = 211 sequences.

(d) To calculate the number of sequences with at least three heads, we can subtract the number of sequences with 0, 1, and 2 heads from the total number of sequences. So, the number of sequences with at least 3 heads is 1,048,576 - (1 + 20 + 190) = 1,048,365 sequences.

Answer 2

There are 1,048,576 possible sequences of heads and tails when flipping a coin 20 times. There are 15,504 sequences with exactly 5 heads. There are 211 sequences with at most 2 heads. There are 1,048,365 sequences with at least three heads.

Step-by-step explanation:

(a) The number of sequences of heads and tails possible when flipping a coin 20 times can be calculated using the formula 2^n, where n is the number of times the coin is flipped. In this case, 2^20 = 1,048,576 possible sequences.

(b) To calculate the number of sequences with exactly 5 heads, we use the binomial coefficient formula. The number of sequences with 5 heads is given by C(20, 5) = 20! / (5! * (20-5)!). Evaluating this expression gives us 15,504 sequences.

(c) To calculate the number of different sequences with at most 2 heads, we sum the number of sequences with 0, 1, and 2 heads. The number of sequences with 0 heads is 1, with 1 head is 20, and with 2 heads is C(20, 2) = 190. Therefore, the total number of sequences with at most 2 heads is 1 + 20 + 190 = 211.

(d) To calculate the number of sequences with at least 3 heads, we need to subtract the number of sequences with 0, 1, and 2 heads from the total number of sequences. Therefore, the number of sequences with at least 3 heads is 1,048,576 - (1 + 20 + 190) = 1,048,365.