Question 41.2: Throwing Dice A die is a cube whose six sides are marked by ......
Throwing Dice
A die is a cube whose six sides are marked by dots; one side has one dot, another two dots and so forth. If you roll a single die, the probability of rolling a 4 is 1/6 because there are six sides and only one has exactly four dots. If you roll two dice simultaneously, what is the probably of rolling a 4?
INTERPRET and ANTICIPATE
This is a classical mechanical problem, and if we knew exactly the initial conditions of the dice, we could use dynamics and kinematics to determine how the dice land. The only reason we can apply probability is that we don’t know exactly how the two dice were tossed. This problem gives us a chance to review probability. The probability is found by determining how many possible ways the two dice may land such that the total is 4 out of all the possible ways the dice may land.
SOLVE
Make a sketch (Fig. 41.5), to find all ways the two dice may land. Imagine one die is blue and the other is red. The sketch shows that if blue die is a 1, the red die may be any value 1 through 6 for six possible combinations. The same reasoning can be used if the blue die is 2 for another six combinations, and so forth.
This sketch (Fig. 41.5) only shows the first three possible values for blue die, but there are six. The sketch shows 18 possible combinations, but that is only half of the total possible combinations.
N_{\text {tot }}=36Next make another sketch (Fig. 41.6) to consider all the ways the two dice may be tossed so they total 4.
From Figure 41.6, we see there are three ways to roll a 4.
N=3
The probability of rolling a 4 is the number of ways to roll a 4 divided by the number of possible ways the dice may land.
\begin{aligned}& P=\frac{N}{N_{ tot }}=\frac{3}{36}=\frac{1}{12} \\& P \approx 8 \%\end{aligned}CHECK and THINK
So the probability of rolling a 4 when you roll just one die is twice as likely as when you roll two dice.
