Fair stock and a lucky dip: mean versus median, and whether a game pays
Last year's school fair ran Monday to Friday and the tuck shop recorded the number of sandwich packs sold each day: 42, 45, 38, 47 and 128. The 128 was Friday, when a regional sports meeting brought visitors onto the compound. This year the committee must order stock for a normal five-day fair and is using last year's figures to decide. One member calculates the average and says 'we should order about 60 packs a day'. Another wants to plan for 128 every day 'to be safe'. The committee is also running a lucky dip: a bag holds 10 counters - 5 red, 3 blue and 2 green. A draw costs $100. A red counter wins $300, a blue wins $150 and a green wins nothing. The counters are replaced after each draw. A parent says the game is a sure earner because 'there are more reds than anything else'.
1. Calculate the mean and the median of the five daily sales. Which one better represents a typical day, and what should the committee order?
Read the worked answer
Mean: (42 + 45 + 38 + 47 + 128) ÷ 5 = 300 ÷ 5 = 60 packs
Median: order the values: 38, 42, 45, 47, 128. The middle value is 45 packs.
The median is the better picture of a typical day. Four of the five days were between 38 and 47; the sports meeting made one day extraordinary. The mean of 60 is pulled upwards by the 128 and is a level the shop reached on only one day out of five.
Ordering: stock for about 45–48 packs a day, plus a separate plan for any event day. Ordering 60 a day means buying 300 packs across the week when ordinary days suggest roughly 217 (42 + 45 + 38 + 47 + 45). At $250 a pack that is about 83 × $250 = $20,750 of stock unsold. The mean is a useful total for the week, but it is the wrong number for deciding each day's order.
2. Evaluate the two members' plans: 60 packs a day and 128 packs a day. Use numbers.
Read the worked answer
Plan A - 60 a day: 5 × 60 = 300 packs. Expected normal-week sales ≈ 42 + 45 + 38 + 47 + 45 = 217 packs. Surplus ≈ 83 packs. At $250 a pack, about $20,750 of stock is at risk of being wasted. The plan is not crazy - it covers a good day - but it is planned around a figure that only one day produced.
Plan B - 128 a day: 5 × 128 = 640 packs, more than double the realistic week. Surplus ≈ 640 − 217 = 423 packs, roughly $105,750 of stock. This is the 'be safe' plan that creates the biggest loss.
Both plans treat one unusual day as normal. The safer approach keeps a normal-day order of about 45, watches the weather and school calendar, and holds a small reserve for a busy day - the 128 tells the committee what a peak looks like, not what a typical day looks like.
3. Is the lucky dip a sure earner? Use expected values to advise the committee, and comment on the parent's reasoning.
Read the worked answer
There are 10 counters and they are replaced, so each colour's chance stays the same.
- Red: 5/10 = 1/2; prize share = 1/2 × $300 = $150
- Blue: 3/10; prize share = 3/10 × $150 = $45
- Green: 2/10; prize = 2/10 × $0 = $0
Expected payout per draw = 150 + 45 + 0 = $195
The stall takes $100 and pays out $195 on average, so it loses about $95 per draw - over 200 draws, roughly $19,000. The game is a sure loser, not a sure earner.
To break even the price must be at least $195; charging $200 would earn about $5 per draw, and 200 draws would raise roughly $1,000. Alternatively the prizes could be cut: making red $150 and blue $75 gives an expected payout of $97.50, under the $100 charge.
The parent has confused 'most likely to be drawn' with 'good for the stall'. Red is the most common counter, but a $300 prize on a 1-in-2 chance costs more than the ticket price - the big prize, not the colour mix, is what sinks the game.