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 Numbered Cards (Posted on 2007-07-23)
Consider a deck of 10 cards numbered in order from 1 to 10. Pick up the first five cards (1 to 5). Randomly discard one and take the 6. Randomly discard one again and take the 7. Continue until the 10 has just been taken.

a) What is the expected average of the five cards in the final hand?
b) What is the expected value of the smallest card in the final hand?

Recompute parts a) and b) where you still hold 5 cards, but go all the way through a 100 card deck.

 See The Solution Submitted by Jer Rating: 4.0000 (1 votes)

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 statistics on the 126 | Comment 12 of 13 |

From the list of results of the 5^5 possible choices of cards to replace, the following numbers of resulting hands (shown without the 10) exist:

`1234 1 1235 1 1236 2 1237 4 1238 8 1239 16 1245 1 1246 2 1247 4 1248 8 1249 16 1256 2 1257 4 1258 8 1259 16 1267 6 1268 12 1269 24 1278 18 1279 36 1289 54 1345 1 1346 2 1347 4 1348 8 1349 16 1356 2 1357 4 1358 8 1359 16 1367 6 1368 12 1369 24 1378 18 1379 36 1389 54 1456 2 1457 4 1458 8 1459 16 1467 6 1468 12 1469 24 1478 18 1479 36 1489 54 1567 6 1568 12 1569 24 1578 18 1579 36 1589 54 1678 24 1679 48 1689 72 1789 96 2345 1 2346 2 2347 4 2348 8 2349 16 2356 2 2357 4 2358 8 2359 16 2367 6 2368 12 2369 24 2378 18 2379 36 2389 54 2456 2 2457 4 2458 8 2459 16 2467 6 2468 12 2469 24 2478 18 2479 36 2489 54 2567 6 2568 12 2569 24 2578 18 2579 36 2589 54 2678 24 2679 48 2689 72 2789 96 3456 2 3457 4 3458 8 3459 16 3467 6 3468 12 3469 24 3478 18 3479 36 3489 54 3567 6 3568 12 3569 24 3578 18 3579 36 3589 54 3678 24 3679 48 3689 72 3789 96 4567 6 4568 12 4569 24 4578 18 4579 36 4589 54 4678 24 4679 48 4689 72 4789 96 5678 24 5679 48 5689 72 5789 96 6789 120 `

Counts of the lowest card are:

` 1             1024  2              781  3              570  4              390  5              240  6              120 `

 Posted by Charlie on 2007-07-24 13:42:56

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