A Game of Dice

Let’s play a game. Take one die and roll it. Remember the value of the roll. Then roll it again. What is the probability that after two rolls ‘4’ appears at least once? Ok, we reason, for the first roll of the die we have a 1/6 chance of a ‘4’ since the die has 6 possible values. And since the roll of the die the second time is not influenced by the value of the die we obtained after the first roll, we assign a 1/6 chance of a ‘4’ for the second roll as well. Now since we have 2 chances of getting a ‘4’ our odds should double then. So we add the two probabilities together. 1/6 + 1/6 = 1/3. We expect a 1/3 probability of seeing at least one ‘4’ after rolling the die twice. Now I ask “What is the probability that after 10 rolls of the die, a ‘4’ appears at least once. Ok, simple, let’s add them up. 1/6 probability for each roll of the die times the number of times we roll the die, i.e. 10 should give us the number. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + /1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 10/6. “10/6 probability of getting a ‘4’”, we proudly say.

Wait a minute! How come you have a probability that exceeds 1? I thought a probability value can only range from 0 (absolutely impossible) to 1 (happens always every time). Obtaining 10/6 or 167% must mean that we are absolutely certain this will happen after 10 rolls and just to make sure we’ve added 67% padding on top for good measure!

Well, something must be wrong in our calculations or thinking. We know a probability of any event or series of events can never exceed 1. Where did we go wrong?