We are sometimes more interested in the probability that an outcome occurs than in the probability of the hypotheses. We then talk about probabilistic predictions.

## Probabilistic Predictions

Probabilistic predictions are when we talk about things like: “It will rain tomorrow with a probability of 60%, and not rain with a probability of 40%”

We often use probabilistic predictions for medical treatment outcomes, sports betting, elections and stock price predictions.

Probabilistic predictions assign each outcome a probability.

**Example:** Suppose we have four coins of three types in a drawer:

- Type A has probability of 0.5 for tails
- Type B has probability of 0.6 for tails
- Type C has probability of 0.9 for tails

In the drawer are two of type A, one of type B and one of Type C. We then have the following Bayesian Updating Table:

Hypothesis | Prior | Likelihood | unnormalised Posterior | normalised Posterior |

H | P(H) | P(D|H) | P(D|H)P(H) | P(H|D) |

A | 0.5 | 0.5 | 0.25 | 0.4 |

B | 0.25 | 0.6 | 0.15 | 0.24 |

C | 0.25 | 0.9 | 0.225 | 0.36 |

Total | 1 | 0.625 | 1 |

What are the prior and posterior probabilistic predictions?

**Answer:** We calculate the probabilistic predictions with the Law of Total Probability. We therefore have for the prior and posterior predictive probabilities:

P(heads) and P(heads|D) are just the complement probabilities of P(tails) and P(tails|D), respectively.

**Remember:** Predictive Probabilities tell us the probabilities for outcomes not for hypotheses.