We throw two conventional dice, and ask the probablity of the following event:
- The main probability $p_m$ to compute is composed of two sub-events: first sub-event, having either of the two dice with a value of 1 or 4, and second sub-event: the two taking values 1 and 4 at the same time.
So I'm trying to compute the probability of either of these two sub-events occuring. My attempt:
- First subevent: for a single die, the probability of either 1 or 4 is 1/6. For two dice, we have 36 outcomes, 6 ways (1,x) can occur, x can be any value, similarly 6 ways (4,x), and because we have two dice, both are multiplied by two, so finally the probablity of subevent one is: $$p_1=\frac{2*6+2*6}{36}\approx 0.66$$
- For the second subevent: there are two ways 1 and 4 can occur at the same time, that is, (1,4) or (4,1) so $p_2 = 2/36 \approx 0.05$
- Finally for the main event, $p_m = p_1+p_2 \approx 0.71.$ Is the estimation correct? Have I gone wrong somewhere with the composition of probabilities?
Note: we assume the two dice are indistinguishable.