Probability of Rolling Dice
The probability of rolling dice, including the likelihood of rolling a 6 once, at least one 6 in two rolls, and at least one 6 in three rolls. Dive into the mathematical analysis of dice outcomes and probabilities.
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Presentation Transcript
Take a break and roll some dice Note - not that we are promoting gambling or anything, but we would appreciate a 10% cut of any winnings as a result of things you learn from Lab 3
If you roll a die ONCE, what is the probability that you will roll a 6?
If you roll a die ONCE, what is the probability that you will roll a 6? 1 2 3 4 5 6 1/6 = 0.17 or about 17%
If you roll a die TWICE, what is the probability that at least one of the rolls will be a 6?
If you roll a die TWICE, what is the probability that at least one of the rolls will be a 6? 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66 11/36 = 0.31 or about 31%
If you roll a die THREE times, what is the probability that at least one of the roles will be a 6?
If you roll a die THREE times, what is the probability that at least one of the roles will be a 6? 111 112 113 114 115 116 211 212 213 214 215 216 311 312 313 314 315 316 121 122 123 124 125 126 221 222 223 224 225 226 321 322 323 324 325 326 131 132 133 134 135 136 231 232 233 234 235 236 331 332 333 334 335 336 141 142 143 144 145 146 241 242 243 244 245 246 341 342 343 344 345 346 151 152 153 154 155 156 251 252 253 254 255 256 351 352 353 354 355 336 161 162 163 164 165 166 261 262 263 264 265 266 361 362 363 364 365 366 411 412 413 414 415 416 511 512 513 514 515 516 611 612 613 614 615 616 421 422 423 424 425 426 521 522 523 524 525 526 621 622 623 624 625 626 431 432 433 434 435 436 531 532 533 534 535 536 631 632 633 634 635 636 441 442 443 444 445 446 541 542 543 544 545 546 641 642 643 644 645 646 451 452 453 454 455 456 551 552 553 554 555 556 651 652 653 654 655 636 461 462 463 464 465 466 561 562 563 564 565 566 661 662 663 664 665 666 91 out of 6 x 6 x 6 = 216 possibilities
If you roll a die THREE times, what is the probability that at least one of the roles will be a 6? 111 112 113 114 115 116 211 212 213 214 215 216 311 312 313 314 315 316 121 122 123 124 125 126 221 222 223 224 225 226 321 322 323 324 325 326 131 132 133 134 135 136 231 232 233 234 235 236 331 332 333 334 335 336 141 142 143 144 145 146 241 242 243 244 245 246 341 342 343 344 345 346 151 152 153 154 155 156 251 252 253 254 255 256 351 352 353 354 355 336 161 162 163 164 165 166 261 262 263 264 265 266 361 362 363 364 365 366 411 412 413 414 415 416 511 512 513 514 515 516 611 612 613 614 615 616 421 422 423 424 425 426 521 522 523 524 525 526 621 622 623 624 625 626 431 432 433 434 435 436 531 532 533 534 535 536 631 632 633 634 635 636 441 442 443 444 445 446 541 542 543 544 545 546 641 642 643 644 645 646 451 452 453 454 455 456 551 552 553 554 555 556 651 652 653 654 655 636 461 462 463 464 465 466 561 562 563 564 565 566 661 662 663 664 665 666 91 / 216 = 0.42 or about 42%
If you roll a die FOUR times, what is the probability that at least one of the roles will be a 6?
If you roll a die FOUR times, what is the probability that at least one of the roles will be a 6? Isn t there a better way? (Yes )
Recall: The probability of something happening is 1 (or 100%) minus the probability of it not happening. (Either it happens, or it doesn t) So ..
If you roll a die ONCE, what is the probability that you will roll a 6? 1/6
If you roll a die ONCE, what is the probability that you will roll a 6? 1/6 If you roll a die ONCE, what is the probability that you will not roll a 6? 1 - 1/6 = 5/6
If you roll a die ONCE, what is the probability that you will roll a 6? 1/6 If you roll a die ONCE, what is the probability that you will not roll a 6? 1 - 1/6 = 5/6 If you roll a die TWICE, what is the probability that you will not roll a 6?
If you roll a die ONCE, what is the probability that you will roll a 6? 1/6 If you roll a die ONCE, what is the probability that you will not roll a 6? 1 - 1/6 = 5/6 If you roll a die TWICE, what is the probability that you will not roll a 6? Two things have to happen: You have to not roll a 6 on the first throw And also not roll a 6 on the second throw
If you roll a die ONCE, what is the probability that you will roll a 6? 1/6 If you roll a die ONCE, what is the probability that you will not roll a 6? 1 - 1/6 = 5/6 If you roll a die TWICE, what is the probability that you will not roll a 6? Two things have to happen: You have to not roll a 6 on the first throw And also not roll a 6 on the second throw And to get the probability of two things happening, you multiply their individual probabilities.
If you roll a die TWICE, what is the probability that at least one roll will be a 6?
If you roll a die TWICE, what is the probability that at least one roll will be a 6? If you roll a die ONCE, what is the probability that you will not roll a 6? 1 - 1/6 = 5/6 If you roll a die TWICE, what is the probability that you will not roll a 6? 5/6 x 5/6 = 25/36 = 0.69 or 69%
If you roll a die TWICE, what is the probability that at least one roll will be a 6? If you roll a die ONCE, what is the probability that you will not roll a 6? 1 - 1/6 = 5/6 If you roll a die TWICE, what is the probability that you will not roll a 6? 5/6 x 5/6 = 25/36 = 0.69 or 69% So .If you roll a die TWICE, what is the probability that you will roll a 6? 1 0.69 = 0.31 or about 31%
Compare to brute force method: 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66 11/36 = 0.31 or about 31%
If you roll a die THREE times, what is the probability that at least one of the rolls WILL be a 6?
If you roll a die THREE times, what is the probability that at least one of the rolls WILL be a 6? If you roll a die THREE times, what is the probability that at least one of the rolls WILL NOT be a 6? 5/6 x 5/6 x 5/6 = (5/6)3 = 125/216 = 0.58 or ~58%
If you roll a die THREE times, what is the probability that at least one of the rolls WILL be a 6? If you roll a die THREE times, what is the probability that at least one of the rolls WILL NOT be a 6? 5/6 x 5/6 x 5/6 = (5/6)3 = 125/216 = 0.58 or ~58% If you roll a die THREE times, what is the probability that at least one of the rolls WILL be a 6? 1 0.58 = 0.42 or ~ 42%
Compare to brute force method: 111 112 113 114 115 116 211 212 213 214 215 216 311 312 313 314 315 316 121 122 123 124 125 126 221 222 223 224 225 226 321 322 323 324 325 326 131 132 133 134 135 136 231 232 233 234 235 236 331 332 333 334 335 336 141 142 143 144 145 146 241 242 243 244 245 246 341 342 343 344 345 346 151 152 153 154 155 156 251 252 253 254 255 256 351 352 353 354 355 336 161 162 163 164 165 166 261 262 263 264 265 266 361 362 363 364 365 366 411 412 413 414 415 416 511 512 513 514 515 516 611 612 613 614 615 616 421 422 423 424 425 426 521 522 523 524 525 526 621 622 623 624 625 626 431 432 433 434 435 436 531 532 533 534 535 536 631 632 633 634 635 636 441 442 443 444 445 446 541 542 543 544 545 546 641 642 643 644 645 646 451 452 453 454 455 456 551 552 553 554 555 556 651 652 653 654 655 636 461 462 463 464 465 466 561 562 563 564 565 566 661 662 663 664 665 666 91 / 216 = 0.42 or about 42%
If you roll a die n times, what is the probability that you will roll at least one 6?
If you roll a die n times, what is the probability that you will roll at least one 6? = 1 (probability of not rolling a 6 in one roll)n or = 1 - (1 - 1/6)n
General formula for determining the probability of something occurring in n times. Think it through and you won t have to memorize it. Given a probability of something happening once ( a single occurrence): Subtract the probability of a single occurrence from 1, to get the single probability of non-occurrence. Raise to the nth power to get the probability of non-occurrence in n times. Subtract from 1 to get the probability of occurrence in n times.
