九州大学 システム情報科学府 情報理工学専攻・電気電子工学専攻 2023年8月実施 確率・統計

Author

Casablanca, 祭音Myyura

Description

箱の中に 個の白いボールと 個の黒いボールがあり, その総数を とする。この箱から つのボールをランダムに選び, 両方が白いボールである確率は であるとする。

(1) が奇数のとき の最小値を求めよ。

(2) が偶数のとき の最小値を求めよ。

(3) を値の小さい順に つ求めよ。

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A box contains white and black balls, and the total number of balls is . When two balls are randomly drawn from the box, the probability that both balls are white is .

(1) Find the minimum value of when is an odd number.

(2) Find the minimum value of when is an even number.

(3) Find the three smallest values of .

Kai

Let denote the event "both balls are white", then we have

Since , we have

(1)

Let . Then we have

from which we have

when , get the minimum value .

(2)

Let . Then we have

from which we get

when , get the minimum value .

(3)

From (i) and (ii) , we easily know that the larger , the larger we have and must be a number of squares.

Let , we have , which implies that is odd.

Let . Then we have . Easy to find that , and are three solutions and the corresponding value of is , and .