Answer:
The answer to the question is;
A. The pattern of outcomes of 0 heads, 1 head, 2 heads ... n heads when arranged in the Pascal's triangle is given by the row of the triangles with the lowest at the edges and highest outcome in the middle potion of the rows in the Pascal's triangle.
B. The number of total possible outcomes in a row n is 2ⁿ.
Explanation:
To solve the question, we note that.
Pascal's triangle is an important sequence of numbers formed by starting with 1 at the top of the triangle followed by 1s on the outer edges of the triangle and the numbers directly below two adjacent numbers is the sum of the two adjacent numbers.
In probability theory, such as in throwing a coin, Pascal's triangle indicates the number of possible heads obtained when n number of coins is thrown, that is, 0 heads, 1 head 2 heads etc. as follows
Number of coins Possible head outcome Total outcome
0 coin 1 1
1 coin 1 1 2
2 coins 1 2 1 4
3 coins 1 3 3 1 8
4 coins 1 4 6 4 1 16
The above triangle means that when
0 coin is flipped, the number of 0 heads is 1
1 coin is flipped, the number of 0 heads is 1 (T) and 1 head is 1 (H)
2 coins are flipped, the number of 0 heads is 1, (TT), 1 head is 2 (TH), (HT) and 2 heads is 1 (HH)
3 coins are flipped 0 heads = 1 (TTT), 1 head = 3, (TTH), (THT), (HTT), the number of 2 heads = 3, (HHT), (HTH), (TTH) and the number of 3 heads = 1, (HHH).
B. The pattern in the number of total possible outcomes is
1, 2, 4, 8, 16 which is =2 raised to the power of the number of coin tossed for a particular row as follows
Number of coin tossed = 0 = 2⁰ = 1 = Total possible outcome
Number of coin tossed = 1 = 2¹ = 2 = Total possible outcomes
Number of coin tossed = 2 = 2² = 4 = Total possible outcomes
Number of coin tossed = 3 = 2³ = 8 = Total possible outcomes
Number of coin tossed = 4 = 2⁴ = 16 = Total possible outcomes
Therefore total number of possible outcome = 2ⁿ where n = number of coins tossed.