At a major track meet, Tara and Emek used recent performance records to estimate their relay team's chance of reaching the final. The data showed probabilities of 0.96, 0.82, and 0.94 for clean first, second, and third baton exchanges. Records also indicated a 0.90 probability that the four runners, as a group, would meet their target running times. Tara and Emek treated the exchange outcomes as approximately independent of one another and, for their model, treated the target-time event as independent of the exchanges. They estimated the probability of meeting all four conditions as 0.96 times 0.82 times 0.94 times 0.90, or about 0.666. Then they examined which factor offered the greatest opportunity for improvement. The second exchange had the lowest success probability at 0.82. If practice raised it to 0.90 while the other probabilities stayed unchanged, the model predicted an overall success probability of about 0.731. Tara cautioned that the independence assumptions were approximations, but the comparison still identified a clear weakness. The team devoted its final practice to the second baton exchange.

Challenges

Challenge 1 of 5

Challenges

Challenge 1 of 4
Hint
Focus on the independence assumption used in the probability model.
Why did Tara and Emek multiply 0.96, 0.82, 0.94, and 0.90 to estimate the probability of meeting all four conditions?
Because the model treats the four conditions as independent, so the probability that all occur is the product of their probabilities.Because probabilities of separate events must always be added before they are compared.Because multiplying the probabilities guarantees that the resulting model is exact.Because the largest probability must be multiplied by the smallest probability to find the overall probability.
Correct. Under the model's independence assumptions, the probability that all four conditions occur is found by multiplying their probabilities.
Not quite. Consider what independence allows you to do when finding the probability that several events all occur.
Next
🎉 Congrats! You got this question!
Hint
Find the difference between the two probabilities, then multiply by 1000.
Using the model values in the story, the original overall success probability is about 0.666 and the improved probability is about 0.731. Approximately how many more successful outcomes would the improved model predict over 1000 similar relay attempts?
Correct. The improved model predicts about 65 additional successful outcomes per 1000 attempts.
Check the difference between 0.731 and 0.666, then scale that difference to 1000 attempts.
Next
🎉 Congrats! You got this question!
Hint
Compare the old and new products by factoring out the probabilities that stayed unchanged.
A student says, "Because the second exchange has the lowest probability, increasing it by 0.08 must increase the overall success probability by exactly 0.08." Which response correctly identifies the mistake?
The claim is correct because changing one factor changes the product by the same amount.The overall increase is smaller because the change of 0.08 is multiplied by the other probabilities in the model.The overall increase must be larger than 0.08 because all probabilities are positive.The overall probability does not change because only one exchange probability was changed.
Correct. The 0.08 improvement is scaled by the other probability factors, so the overall increase is less than 0.08.
Not quite. The overall model is a product, so changing one factor affects the result through multiplication by the remaining factors.
Next
🎉🎉🎉🎉🎉🎉🎉🎉
🎉 Congrats! You finished this challenge set!
Bayrak Yarışı Sonucunu Tahmin Etmek
TR
ARC_G12_DP_02_STORY_03
Bayrak Yarışı Sonucunu Tahmin Etmek
Forecasting the Relay Result
EN
ARC_G12_DP_02_STORY_03
Forecasting the Relay Result