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確率の解法 | 夢を叶える塾
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確率の問題の解法

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ある事象が全体の事象のうちどれほど起こりうるのかを求める確率の問題です。

基本問題

(1)2つのサイコロを振って、出た目がぞろ目となる確率をいくらか
(2)赤玉が4つと白玉が3つの7つの玉が入っている袋から2つ取り出したとき、赤玉が2つとなる確率はいくらか

解き方

確率の基本は該当事象を全事象で割ることです。

解説

(1)2つのサイコロを振って、出た目がぞろ目となる確率をいくらか
確率の問題なので、該当事象と全事象それぞれの場合の数を求める必要があります。

まず全事象を求めましょう。
これは、2つのサイコロの出る目の通り数ですね。
6\times 6=36で36通りになります。

次に該当事象を求めましょう。
2つのサイコロの出る目がぞろ目の通り数ですね。
数えるまでもないかもしれませんが、
(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)
6通りになります。

確率は、\displaystyle \frac{6}{36}=\frac{1}{6}になります。

(2)赤玉が4つと白玉が3つの7つの玉が入っている袋から2つ取り出したとき、赤玉が2つとなる確率はいくらか
(1)同様、全事象と該当事象の場合の数をそれぞれ求めていきます。

まずは全事象を求めましょう。
7つの玉から2つ選ぶわけですね。
\displaystyle \frac{7\times 6}{2\times 1}=21で21通りになりました。

次に赤が2つという該当事象を求めましょう。
4つの赤玉から2つ選べれば良いですね。
\displaystyle \frac{4\times 3}{2\times 1}=6で6通りになりました。

確率は、\displaystyle \frac{6}{21}=\frac{2}{7}になります。

応用問題

(1)基本問題の(2)で袋から1つ取り出して玉を戻しもう一度取り出したとき、玉が同じ色になる確率はいくらか
(2)基本問題の(2)で袋から1つ取り出して玉を戻しもう一度取り出したとき、玉が異なる色になる確率はいくらか

解き方

応用問題も基本問題の積み重ねになります。
一つ一つ確認していきましょう。

解説

(1)基本問題の(2)で袋から1つ取り出して玉を戻しもう一度取り出したとき、玉が同じ色になる確率はいくらか
少し難しくなっていますね。
しかし基本は変わりません。
まず全事象から求めていきましょう。

7つから1つを取り出す、という事を2回行います。
つまり、\displaystyle \frac{7}{1}=7が2回分で、
7\times 7=49
全事象は49通りあります。

次に玉が同じ色になる該当事象です。
これは赤が2つという場合と、白が2つという場合の2通りがありますね。
それぞれの事象は異なる事象なのでそれぞれの事象の通り数の和が通り数になります。

赤2つは4通りが2回ですから、4\times 4=16の16通りです。
白2つは3通りが2回ですから、3\times 3=9の9通りです。
合計25通りが該当事象です。

確率は、\displaystyle \frac{25}{49}になります。

(2)基本問題の(2)で袋から1つ取り出して玉を戻しもう一度取り出したとき、玉が異なる色になる確率はいくらか
これも(1)と同様の解き方で解くことができます。
しかし、(1)の全く逆のパターンになっています。
こういう場合は余事象を考えても答えを導くことができます

確率は、\displaystyle 1-\frac{25}{49}=\frac{24}{49}になります。

終わりに

確率は中学生の数学の中では他と関連が薄いという点で少し特殊かもしれません。
しかし高校入試問題の計算問題で出題されています。
また、高校生の数学でもまた場合の数と確率が出てきます。
中学生の確率はそのベースにもなります。

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