Two players take turns rolling a fair die. Player A wins if he rolls a 6, Player B wins if he rolls a 5 or 6. If A starts, find the probability that A wins.
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Step-by-Step Solution
Step 1: Define probabilities for each player
First, we determine the probability of each player winning on their turn. Player A wins if they roll a 6, so the probability is 1/6. Player B wins if they roll a 5 or a 6, so the probability is 2/6=1/3.
Step 2: Define probabilities of not winning
Next, we calculate the probability that each player does not win on their turn. For Player A, this is 1−1/6=5/6. For Player B, this is 1−1/3=2/3. These probabilities are crucial for considering turns where neither player wins.
Step 3: Formulate the probability of A winning
Player A can win on their first turn, or on their second turn (if both A and B fail on their first turns), or on their third turn (if both A and B fail on their first two turns), and so on. This forms an infinite geometric series.
Step 4: Write the geometric series
Substituting the probabilities, the series becomes 1/6+(5/6)(2/3)(1/6)+(5/6)(2/3)(5/6)(2/3)(1/6)+…. This is a geometric series with first term a=1/6 and common ratio r=(5/6)(2/3).
Step 5: Calculate the common ratio
The common ratio r is the probability that neither player wins in a round (A's turn then B's turn). This is calculated as P(Anot win)×P(Bnot win)=(5/6)×(2/3)=10/18=5/9.
Step 6: Apply the sum of an infinite geometric series formula
For an infinite geometric series with ∣r∣<1, the sum is given by S=a/(1−r). Here, a=1/6 and r=5/9.
Step 7: Calculate the final probability
Substitute the values into the formula: P(Awins)=(1/6)/(1−5/9)=(1/6)/(4/9). This simplifies to (1/6)×(9/4)=9/24=3/8.