antiphon

For two events and , if then we say that and [/url'] are disjoint. Does disjointedness differ to mutual exclusiveness? Are they both the same or are they different?

They are the same.

The term "disjoint" is quite common in measure theory, where arbitrary mathematical sets are considered. "Mutually exclusive" is a heuristic phrase that has no meaning in the case of general measures. Probability measure (i.e. measure with[math]P(\Omega)=1[/math]) gives the sets [math]A\subseteq\Omega[/math] the interpretation of "events", so saying "mutually exclusive events" makes sense.

Cheers,

Tuomas

Looks like there are multiple ways to understand this... let me present the usual way (at least for me) of solving elementary examples of conditional probabilities:

Let [math]S[/math] denote the event that father scores, and let [math]R[/math] denote the event that the son reports scoring. The given probabilities are [math]P(S)=0.6[/math] and [math]P(R | S)=0.8[/math], and from the formulation of the problem it follows that [math]P( \bar R | \bar S)=0.8[/math] and [math]P(R | \bar S)=0.2[/math] (where [math]\bar A[/math] denotes the complement of [math]A[/math]).

Now we can use Bayes formula + law of total probability to compute what we want:

[math]P(S | R)=\frac{P(S \cap R)}{P®}=[/math]

[math]=\frac{P(R|S) P(S)}{P(R|S)P(S)+P(R|\bar S)P(\bar S)}=\frac{0.8\cdot0.6}{0.8\cdot0.6+0.2\cdot 0.4}\approx 0.857 [/math].

Notice that the probability goes up from 0.6 because we add information: this is the main idea of Bayesian thinking. A priori (before information from the son) we have more uncertainty about scoring, and afterwards, a posteriori, we are more certain.

Cheers,

Tuomas