Compute 𝐄(𝑋) for the following random variable 𝑋:
𝑋=Number of tosses until getting 4 (including the last toss) by tossing a fair 10-sided die.
𝐄(𝑋)=
Compute the probability of rolling an odd number when tossing a fair 10-sided die.
Probability of rolling an odd number=
To compute 𝐄(𝑋), the expected value of the random variable 𝑋, we need to find the average number of tosses until getting a 4.
To determine this, we can consider that the probability of getting a 4 on any given toss is 1/10, since we are assuming the die is fair.
Let's define 𝑋₀ as the number of tosses until getting a 4 for the first time. The probability of getting a 4 on the first toss is 1/10, so 𝑋₀ = 1.
Now, for 𝑋₁ (the number of tosses until getting a 4, starting from the second toss), we have two possibilities: either we get a 4 on the second toss, with a probability of (9/10) * (1/10), or we roll a number that is not 4 and reset the count to 1. In the second case, we return to the starting position and 𝑋₁ = 𝑋₀ + 1.
By continuing this process, we can derive a recursive formula for 𝐄(𝑋). Let 𝐄(𝑋) represent the expected value of 𝑋:
𝐄(𝑋) = (1/10) * 1 + (9/10) * (1 + 𝐄(𝑋))
Simplifying the equation, we have:
𝐄(𝑋) = 1 + (9/10) * 𝐄(𝑋)
Multiplying both sides by 10, we get:
10𝐄(𝑋) = 10 + 9𝐄(𝑋)
Subtracting 9𝐄(𝑋) from both sides, we obtain:
𝐄(𝑋) = 10
Therefore, 𝐄(𝑋) = 10, meaning that on average, it takes 10 tosses to get a 4 when rolling a fair 10-sided die.
Now, let's compute the probability of rolling an odd number when tossing a fair 10-sided die:
Since the 10-sided die has 10 equally likely outcomes (numbers 1 to 10), and half of these outcomes (numbers 1, 3, 5, 7, 9) are odd, the probability of rolling an odd number is 5/10 or 0.5.
Therefore, the probability of rolling an odd number when tossing a fair 10-sided die is 0.5 or 50%.
To compute 𝐄(𝑋), we need to find the expected value of the random variable 𝑋, which is the average number of tosses until getting a 4.
𝐄(𝑋) can be calculated using the formula:
𝐄(𝑋) = 1/𝑝,
where 𝑝 is the probability of getting a success on each toss. In this case, a success is rolling a 4.
Since we are rolling a fair 10-sided die, the probability of rolling a 4 on a single toss is 1/10.
Therefore, 𝐄(𝑋) = 1 / (1/10) = 10.
Hence, the expected number of tosses until getting a 4 (including the last toss) is 10.
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To compute the probability of rolling an odd number when tossing a fair 10-sided die, we need to determine the number of favorable outcomes (odd numbers) and divide it by the total number of possible outcomes.
A fair 10-sided die has 10 possible outcomes: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
Out of these 10 possible outcomes, 5 are odd numbers: 1, 3, 5, 7, and 9.
Therefore, the probability of rolling an odd number is 5/10 or 0.5.
Hence, the probability of rolling an odd number when tossing a fair 10-sided die is 0.5 or 50%.