Joint Event Probabilities

The probability of a ‘joint’ event – e.g. drawing a white ball followed by two black balls (from a jar containing 7 white and 5 black balls) can be calculated using total sample space.

Instead of calculating the individual probabilities of drawing a white ball – and then a black ball and then again a black ball, it is easier to just calculate the total number of outcomes (total Sample Space).

# of possible events (# ways in which the event can occur) / Total Sample Space  = probability

So – in our example – the total sample space = 12 balls – and three withdrawals – i.e. – 12 * 11 * 10 is the TOTAL number of ways to draw out three balls. This is part 1 of the calculation.
Now – total number of ways of drawing out 1 white ball (on the first try) – is 7. Total Number of ways of drawing out 2 black balls on the next two tries is 5 * 4.

However – these three draws can be interchanged – i.e. the first one can be black, the second white then the third black. So there’s THREE  permutations of the above that lead to the same result – 1 white and 2 black.

Therefore total number of ways these three balls can be drawn out = 3 * 7 * 5 * 4 = 420

Total Sample Space = 1320

Probability = 420/1320

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