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Numbered Cards (Posted on 2007-07-23) Difficulty: 4 of 5
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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