Question

Suppose Elizabeth Warren has a 66% chance of becoming the Democratic Party candidate for President, and Kamala Harris has a 34% chance of becoming the Democratic Party candidate in 2020. If Donald Trump runs against Elizabeth Warren, he has a 40% chance of winning the election. If Donald Trump runs against Kamala Harris, he has a 35% chance of winning the election. If Trump loses the election, what is the probability that he ran against Elizabeth Warren?

Answer 1

Elizabeth Warren has a 66% chance of becoming the Democratic Party candidate for President

Probability that Elizabeth Warren becomes the Democratic Party candidate for President = P(E₁) = 66/100 = 0.66

Kamala Harris has a 34% chance of becoming the Democratic Party candidate for president

Probability that Kamala Harris becomes the Democratic Party candidate for President = P(E₂) = 34/100 =0.34

If Donald Trump runs against Elizabeth Warren, he has a 40% chance of winning the election which implies that he has a 60% chance of losing the election to Elizabeth Warren

Let L denote the event that Trump loses the election.

Then, Probability that Trump loses the election given that he contests against Elizabeth Warren = P(L|E₁) = 60/100 = 0.6

If Donald Trump runs against Kamala Harris, he has a 35% chance of winning the election which implies that he has a 65% chance of losing the election to Kamala Harris

Probability that Trump loses the election, given that he contests against Kamala Harris = P(L|E₂) = 65/100 = 0.65

This question can be solved using Baye's Law which states that

If E₁, E₂, .............., E_n are n-mutually exclusive events with P(E_i) ≠ 0, for i = 1, 2, 3,.................., n. For any arbitrary event A which is a subset of such that P(A) > 0, then

If Trump loses the election, what is the probability that he ran against Elizabeth Warren?

This probability is represented as P(E₁|L) and is calculated below

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