My Math Forum Longest hit streak in baseball

 December 17th, 2015, 11:58 AM #1 Newbie   Joined: Dec 2015 From: Kansas Posts: 2 Thanks: 0 Longest hit streak in baseball A batter hits the ball 20% of the time at bat. The batter gets three times at bat in each game, and there are 162 games each year. In a year, what would you expect the longest streak of consecutive games in which the batter hits at least one hit in each game? Assume that hits and games are independent. I forgot how to do this type of problem any help is greatly appreciated
 December 17th, 2015, 04:05 PM #2 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 661 Thanks: 87 The probability of at least 1 hit in any one game is 1 - (1-0.2)^3 = 0.488. Although I don't know how to solve the problem, I generated 162 random numbers from 1 to 1,000 at Free Online Random Number Generator and Checker and here is my first set of 162 numbers. 608 476 539 33 170 979 882 284 702 173 898 954 481 438 298 871 196 14 26 540 251 348 584 332 131 709 365 401 221 664 554 183 966 871 772 729 825 438 594 141 729 745 106 872 410 558 362 827 518 425 853 931 835 264 543 642 245 232 453 268 496 504 447 553 357 382 798 886 742 96 188 865 444 904 742 234 348 108 664 872 625 664 703 442 959 766 7 255 426 692 989 284 211 705 610 189 726 823 343 643 624 704 1 287 360 281 474 618 850 881 211 118 929 822 475 694 636 335 521 500 405 415 964 302 893 154 297 241 658 471 724 407 175 503 398 140 533 421 203 625 580 462 62 327 777 466 944 149 611 575 114 660 758 400 118 417 274 8 495 826 76 845 You can count the longest streak of consecutive numbers from 1 to 488 (representing games with at least 1 hit) on your own and do more simulations if you want. I divided the 162 random numbers into 18 rows of 9. A hitting streak can continue from one row to the next.
 December 31st, 2015, 06:19 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 If the batter has a 20%= 0.02= 1/5 probability of getting a hit at each "at bat" then there is a 80%= 0.08= 4/5 probability that he will NOT get a hit. The probability that he does NOT get a hit in three "at bats" is (0.(0.(0.= 0.512 so the probability he does get at least one hit in a game is 1- 0.512= 0.488. The probability he gets at least one hit in n consecutive games is $0.488^n$. The expected number or consecutive games in which he gets a hit is $\sum_{n=0}^\infty n0.488^n$. Except for the 'n' multiplied, that is a geometric series. We can note that "$n r^n= (n+1)r^n- r^n$ so $\sum nr^n= \sum (n+1)r^n- \sum r^n$. The second of those is a geometric series while the second is the derivative of the sum $\sum r^{n+1}$ which is a geometric series except that it is missing the first term. The sum of $\sum_{n=0}^\infty r^n$ is $\frac{1}{1- r}$ (as long as |r|< 1) and the sum of $\sum{n=0}^\infty r^{n+1}$ is $\frac{1}{1- r}- 1$ which has derivative, with respect to r, $\frac{1}{(1- r)^2$. So the sum here is $\frac{1}{(1- r)^2}- \frac{1}{1- r}$. Evaluate that with r= 0.0488.

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