問26
頻出度★★★
X 社では来期に製品 A, B, C, D のいずれか一つを市場に投入することを検討している。来期の市況で好況の確率は 40%, 不況の確率は 60%と予想され, 市況ごとの各製品の予想売上高が表のとおりであるとき, この市場から得られる最大の売上期待値をもたらす製品はどれか。 単位 百万円
| 市況 | 製品A | 製品B | 製品C | 製品D |
|---|---|---|---|---|
| 好況 | 12 | 18 | 20 | 10 |
| 不況 | 10 | 8 | 6 | 12 |
平成23年度 特別試験 ITパスポート試験 問26
ストラテジ系企業活動 › 業務分析・データ利活用
This question's explanation and your answers so far.
解説
問 26
Incorrectア
The expected sales of Product A are 12 × 0.4 + 10 × 0.6 = 4.8 + 6.0 = 10.8 million yen, the smallest of the four products. Product A gives 12 million yen in a boom and 10 million yen in a slump, so it is stable with only a small swing between market conditions, but its absolute figures are not high under either condition, so its expected value does not rise either. A small swing (low risk) and a large expected value are two different things, and what this question asks about is strictly the size of the expected value, so Product A is not the correct answer.
- イ
正解 The correct answer is イ (Product B). An expected value is the value obtained by multiplying the value in each possible state by the probability that the state occurs, and then adding all of them together. In this question the market has only two states, a boom (probability 40% = 0.4) and a slump (probability 60% = 0.6), so the expected sales of each product can be found with the following formula: expected sales = sales in a boom × 0.4 + sales in a slump × 0.6. Substituting the figures from the table into this formula and calculating all four products gives: Product A = 12 × 0.4 + 10 × 0.6 = 4.8 + 6.0 = 10.8 million yen, Product B = 18 × 0.4 + 8 × 0.6 = 7.2 + 4.8 = 12.0 million yen, Product C = 20 × 0.4 + 6 × 0.6 = 8.0 + 3.6 = 11.6 million yen, and Product D = 10 × 0.4 + 12 × 0.6 = 4.0 + 7.2 = 11.2 million yen. Comparing the four, 12.0 > 11.6 > 11.2 > 10.8, so the product that yields the largest expected sales is Product B. What deserves attention here is that the largest sales in a boom belong to Product C (20 million yen) and the strongest figure in a slump belongs to Product D (12 million yen); choosing on the basis of only one of those rows leads to a wrong answer. Because the probability of a slump is 60%, higher than that of a boom, the slump figures have a greater effect on the expected value. Product C is the largest in a boom, but its slump figure of 6 million yen is the lowest of the four, and because it sinks on the side carrying the greater probability weight, its expected value does not reach Product B. Product B, by contrast, has the second-highest boom figure at 18 million yen, and its slump figure of 8 million yen is not extremely low either, so it comes first in the weighted total of both. As this shows, the correct way to solve an expected-value question is always to calculate the probability-weighted total for every choice and then compare them.
Incorrectウ
The expected sales of Product C are 20 × 0.4 + 6 × 0.6 = 8.0 + 3.6 = 11.6 million yen, which falls short of Product B's 12.0 million yen and ranks second. Choosing it just because of the 20 million yen in a boom, the largest single figure in the table, is a typical wrong-answer pattern; but the probability of a boom is only 40%, while in the slump that accounts for 60% of the probability its sales are 6 million yen, the lowest of the four products, so it is overtaken once the figures are weighted and summed. The judgment must be based on the total after multiplying by the probabilities, not on the size of one cell of the table.
Incorrectエ
The expected sales of Product D are 10 × 0.4 + 12 × 0.6 = 4.0 + 7.2 = 11.2 million yen, below Product B's 12.0 million yen and ranking third. Product D is the strongest of the four products in a slump at 12 million yen, and as a choice that emphasizes the slump side with its higher probability it may look reasonable. However, its boom figure of 10 million yen is the lowest, so it is left far behind on the 40% side, and its total still does not match Product B. Choosing by looking only at the side with the higher probability is likewise the error of looking at only one side, the same point as in the case of Product C.
ここがポイント
- Calculating an expected value — expected value = Σ (the value obtained in each state × the probability that the state occurs). In this question, that means calculating sales in a boom × 0.4 + sales in a slump × 0.6 for each choice and picking the largest. It is a good idea to confirm that the probabilities add up to 1 (40% + 60% = 100%) before starting the calculation.
- How to read a decision table (payoff table) — In a table whose rows represent states such as market conditions and whose columns represent the alternatives (products), you must not draw a conclusion from the size of a single cell, or from a one-sided comparison that looks only at the boom row or only at the slump row. Always work out the probability-weighted total for every alternative and compare them. Also keep in mind that the higher the probability of a state, the greater the influence its figures have on the expected value.
用語