Member: reneau34

Hello

reneau34Feb 21, 2023

To solve this problem, we can start by listing all the possible outcomes of flipping a fair coin two times:

HH (two heads)

HT (one head, one tail)

TH (one tail, one head)

TT (two tails)

We can then calculate the length of the longest run for each outcome:

HH: The longest run is 2.

HT: The longest run is 1.

TH: The longest run is 1.

TT: The longest run is 2.

Therefore, the expected length of the longest run is the average of the longest runs for each outcome:

Expected longest run = (2 + 1 + 1 + 2) / 4 = 1.5

So the expected length of the longest run when flipping a fair coin two times is 1.5.

reneau34Feb 20, 2023

Hello

We can start by using the identity (x + y + z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz

to expand (ab + ac + bc)^3:

(ab + ac + bc)^3 = a^3b^3 + a^3c^3 + b^3c^3 + 3a^2b^2c(a + b + c) + 3a^2bc^2(a + b + c) + 3ab^2c^2(a + b + c) + 6a^2b^2c^2

Notice that we can simplify the terms involving a + b + c by factoring out the common factor:

(ab + ac + bc)^3 = a^3b^3 + a^3c^3 + b^3c^3 + 3abc(a + b + c)(a^2 + b^2 + c^2) + 6a^2b^2c^2

Substituting a = 1, b = 1, and c = -1,

we get:

(ab + ac + bc)^3 - a^3b^3 = 1 + 1 + 1 + 3(1)(1 - 1)(1^2 + 1^2 + (-1)^2) + 6(1)(1)(-1) - 1 = 3

Therefore,

(ab + ac + bc)^3 - a^3b^3 = 3.

hope so you got your answer please give your feedback.

reneau34Feb 17, 2023