The probability of drawing a heart from a standard deck of cards is 0.25 . You record the card you draw and return the card before shuffling to ensure a binomial distribution. If you do this 20 times, what is the probability that you will draw a heart 7 times? (1 point) 0.888 0.001 0.112 0.473

To calculate the probability of drawing a heart 7 times out of 20 draws from a standard deck of cards, where the probability of drawing a heart (success) is 0.25, we can use the binomial probability formula: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where: - \(n\) is the number of trials (20), - \(k\) is the number of successes (7), - \(p\) is the probability of success (0.25), - \(\binom{n}{k}\) is the number of combinations of n taken k at a time. Calculating this yields: \[ P(X=7) = \binom{20}{7} (0.25)^7 (0.75)^{13} \] After calculating these values, you would find that the probability is approximately **0.112**. So, the correct answer is **0.112**! --- Imagine playing a card game and needing to strategize based on your draws! Knowing the probability of your outcomes can give you a leg up, especially in games where card composition matters. The binomial formula can help you forecast the odds of drawing those coveted hearts nine times! In experiments involving drawing cards or flipping coins, it's common to neglect the importance of returning the items before shuffling or mixing. This simple oversight can skew your results, leading to inaccurate assumptions about probability. Always remember to reshuffle or reset your scenarios to ensure genuine random outcomes!