John Has A Choice Between Two Options To Play Tennis Sets

John Has A Choice Between Two Options To Play 3 Tennis Sets With H

John is faced with a decision regarding the sequence of playing tennis sets with his father (F) and Bill (B), a tennis champion. The options are to play in the order F-B-F or B-F-B. Bill is a better player than John’s father. The goal for John is to win a prize, which he will achieve if he wins two consecutive sets. The questions are firstly: which order should John choose to maximize his chances of winning two sets in a row, and why? Secondly, if John instead aims to maximize the expected number of wins, E[Z], should he choose a different order? And what is the reasoning behind this? Additionally, the problem touches on a separate statistical question about binomial random variables X and Y, with parameters (n, p) and (n, q) respectively, and whether the probabilities P(X=k) are less than or equal to P(Y=k) for all k when 0

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John’s decision-making process in choosing the sequence of tennis matches against his father and Bill is a classic problem involving probability and strategic choice. The key factors include the players’ abilities, modeled as probabilities of winning each match, and the specific goal of either consecutively winning two matches to secure a prize or maximizing the expected total number of wins. This analysis involves understanding conditional probabilities, optimal strategies, and the properties of binomial distributions, providing insight into game theory and statistical comparison.

Part 1: Optimal Play Sequence for Consecutive Wins

The primary objective for John is to determine which sequence—F-B-F or B-F-B—maximizes his probability of winning two consecutive sets. We know that Bill (B) is a better player than his father (F), implying that John’s probability of beating Bill (p_B) is less than his probability of beating F (p_F). These probabilities satisfy p_F

In the sequence F-B-F, John faces his father first, then Bill, then his father again. The critical point is whether the initial win influences subsequent probabilities, given the order and potential conditional dependencies. Typically, in such independent match setups, the order does not affect the probability of winning specific matches but impacts the likelihood of achieving two consecutive wins over the three matches.

Analyzing the F-B-F sequence, John wins two consecutive sets if he wins either the first two matches or the last two matches. The probability that John wins the first two matches (F then B) is p_F p_B. The probability that he wins the last two matches (B then F) is p_B p_F. The probability that he wins all three matches in a row (F, B, F) relevant for getting two consecutive wins at some point is computed accordingly, considering overlaps.

Similarly, for the B-F-B sequence, the potential for two consecutive wins occurs if John wins either the first two matches (B then F) with probability p_B p_F, or the last two matches (F then B), which is p_F p_B. Since p_F

After thorough analysis, the optimal sequence for maximizing the chance of two consecutive wins turns out to favor the arrangement where John is more likely to win the initial match that sets the tone. Because Bill is a stronger player (p_B > p_F), starting with F provides a higher chance of creating the sequence of wins needed, followed by leveraging the slightly lower chance of losing subsequent matches.

Therefore, John should choose the sequence F-B-F to maximize his probability of winning two sets in a row, as this arrangement gives him a higher likelihood of starting successfully against his father and then capitalizing on that momentum.

Part 2: Strategy for Maximizing the Expected Number of Wins, E[Z]

Now, consider that John’s goal shifts from winning two consecutive sets to maximizing the expected total number of wins, E[Z], where Z is the total number of sets won in the three matches. The expected total wins depend on the individual probabilities of winning each match, regardless of whether the wins are consecutive.

In this case, the order of play becomes less critical because the total expected number of wins is simply the sum of the individual probabilities of winning each match. Specifically, E[Z] = p_F + p_B + p_F in the F-B-F sequence and is similar in the B-F-B sequence, adjusted for which matches involve playing against F or B. As the probabilities remain the same, the total expected number of wins does not significantly depend on the sequence but more on the per-match winning probabilities.

However, if John aims to maximize the overall expected number of wins, he should prioritize playing matches where his probability of winning is highest, regardless of order. Given p_F

Thus, for maximizing E[Z], the sequence choice is less impactful than the matchups themselves. Consequently, John’s optimal strategy might be simply to play against the weaker opponent F more often if possible, or to consider psychological factors influencing his win probabilities.

Statistical Comparison of Binomial Probabilities

The second part of the question involves comparing two binomial random variables, X ~ Bin(n, p) and Y ~ Bin(n, q), with 0

Recall the probability mass function of a binomial distribution:

P(X = k) = C(n, k) p^k (1 - p)^{n - k}

and similarly for Y:

P(Y = k) = C(n, k) q^k (1 - q)^{n - k}

Since C(n, k) is common to both expressions, the comparison reduces to whether q^k (1 - q)^{n - k} ≥ p^k (1 - p)^{n - k} for all k.

To determine if P(X=k) ≤ P(Y=k) for all k, consider the ratio:

R(k) = P(Y = k) / P(X = k) = [q / p]^k * [(1 - q) / (1 - p)]^{n - k}

Because 0 1, and (1 - q) / (1 - p) p implies 1 - q

Therefore, R(k) increases exponentially with k due to [q / p]^k, but is multiplied by a decreasing factor [(1 - q)/(1 - p)]^{n - k}. The overall behavior depends on the relative sizes of these factors, but generally, R(k) > 1 for larger values of k, and R(k)

Thus, the inequality P(X=k) ≤ P(Y=k) for all k does not hold universally because at small k, P(X=k) could be larger, and at large k, P(Y=k) could dominate. A counterexample with small n demonstrates this: for n=2, p=0.2, q=0.8, calculations show divergence of the probabilities at different k values.

In conclusion, the statement is false; P(X=k) ≤ P(Y=k) for all k does not hold when P(X) is Bin(n, p) and P(Y) is Bin(n, q) with 0

Conclusion

Strategic decision-making in sports and statistical properties of probabilities offer insights into optimal choices and comparative analyses. In the tennis scenario, starting with his father (F) maximizes John’s chance of winning two consecutive sets due to the relative strengths of the opponents. When considering total expected wins, the order is less influential than individual match probabilities. The comparison of binomial probabilities further illustrates the nuanced relationships between parameters p and q, demonstrating that higher success probability p does not guarantee uniformly higher probabilities across all outcomes compared to q.

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