10. A pair of dice is tossed 180 times. Determine the probability that a sum of 7 appears i) Exactly 40 times (Ans: 0.0108 ) ii) Between 25 and 35 inclusive times

When dealing with probabilities involving dice, it’s crucial to understand the likelihood of rolling certain sums. For a pair of dice, there are 6 possible combinations that lead to a sum of 7, out of a total of 36 combinations. This gives us a probability of \( \frac{6}{36} = \frac{1}{6} \). Therefore, when tossed 180 times, we can consider this scenario as a binomial distribution. To find the probability of rolling a sum of 7 exactly 40 times, we can use the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the total number of trials (180), \( k \) is the number of successes (40), and \( p \) is the probability of success (1/6). Plugging in the numbers, we get: \[ P(X = 40) = \binom{180}{40} (1/6)^{40} (5/6)^{140} \approx 0.0108 \] For rolling a sum of 7 between 25 and 35 times, we can sum up the probabilities from 25 to 35 using the same binomial formula for each k value in that range. This range requires computing several probabilities, which can be simplified using normal approximation due to the large number of trials. The mean would be \( n \times p = 180 \times \frac{1}{6} = 30 \) and the standard deviation \( \sqrt{n \times p \times (1-p)} \approx 6.68 \). Using a normal distribution, you can then find the cumulative probabilities for 25 and 35 and subtract them to get the final probability.