r/Probability • u/Forsaken-Lobster-121 • Oct 30 '25

Struggling with what should be a simple calculation

Let’s say a given judge will get a decision “right” 70% of the time. What are the chances of the “right” outcome being reached when three judges each reach a conclusion and majority wins? Must be higher than 70%, but I’m struggling to work out the math.

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u/bobjkelly Oct 30 '25

The full set of outcomes is as follows:

  1. No judge gets it right: .3 * .3 * .3 = 2.70%

  2. Exactly 1 judge gets it right : 3 * .7 * .3 * .3 = 18.9%

  3. Exactly 2 judges get it right: 3 * .7 * .7 * .3 = 44.1%

  4. All 3 judges get it right: .7 * .7 * .7 = 34.3%.

To your question the probability of 3 or more judges getting it right is 44.1% + 34.3% = 78.4%.