a fair dice is tossed seven times. suppose that on tosses 2 through 7, the value seen is different from the value seen on the preceding toss. Let S denote the sum of the 7 value seen, the probability that s to be even is
To find the probability that the sum S is even, we need to consider the possible outcomes that satisfy this condition divided by the total number of possible outcomes.
Let's break down the problem into steps to find the probability:
Step 1: Determine the possible outcomes that satisfy the given condition.
We are tossing a fair dice seven times, and on tosses 2 through 7, the value seen is different from the value seen on the preceding toss. For each toss, there are 6 possible outcomes (numbers 1 through 6). We need to consider two cases:
a) The first toss is an odd number.
b) The first toss is an even number.
Case a) The first toss is an odd number:
In this case, the second toss can be any even number (3 possible outcomes: 2, 4, or 6). From the third toss onwards, the value seen must be different from the preceding toss, which means there are 5 possible outcomes for each toss.
So, the total number of possible outcomes for case a) is: 1 * 3 * 5 * 5 * 5 * 5 * 5 = 3,125.
Case b) The first toss is an even number:
In this case, the second toss can be any odd number (3 possible outcomes: 1, 3, or 5). From the third toss onwards, the value seen must be different from the preceding toss, which means there are 5 possible outcomes for each toss.
So, the total number of possible outcomes for case b) is: 1 * 3 * 5 * 5 * 5 * 5 * 5 = 3,125.
Step 2: Determine the total number of possible outcomes.
For each dice toss, there are 6 possible outcomes. Since we are tossing the dice 7 times, the total number of possible outcomes is: 6^7 = 279,936.
Step 3: Calculate the probability.
The probability that the sum S is even is given by:
P(S is even) = (Possible outcomes that satisfy the condition) / (Total possible outcomes)
= (Possible outcomes for case a) + Possible outcomes for case b)) / (Total possible outcomes)
= (3,125 + 3,125) / 279,936
= 6,250 / 279,936
= approximately 0.0223 (rounded to 4 decimal places)
Therefore, the probability that the sum S is even is approximately 0.0223, or 2.23%.