Expected value for paid random items: what a rare drop costs on average

When a game sells random items, the published rate tells you how likely a single opening is to give you something rare. It doesn't tell you what that rare item costs. For that you need one more number, the price of an opening, and one sum. This guide goes through that sum with real figures, then shows where it stops working. (Whether these items are allowed in the UK, and how PEGI labels them, is covered in loot boxes and UK law.)

The sum

Average cost of a rare item = average number of openings × price of one opening

The average number of openings comes straight from the rate p:

• With no guarantee: 1 ÷ p
• With a guarantee on opening N (pity): (1 − (1 − p)^N) ÷ p

This is the same question as "how much does a 5-star cost on average" in a gacha game. Swap in the 5-star rate, the guarantee and the price of one pull. Our guide to how pity works explains the second formula.

People often call this the expected value of a loot box. Strictly, expected value adds up everything a box could give you, weighted by its chance. That only works if every result has a price, and most in-game items can't be sold or valued in pounds. What you can work out is the expected cost: how much you spend, on average, to get the thing you want.

Worked example: Apex Legends

Everything here comes from EA and the PlayStation Store UK, checked on 29 September 2026.

The rate. EA gives a 7.4% chance of at least one Legendary item in an Apex Pack, and states that a player cannot open more than 30 packs without one.

The price of one opening. EA's FAQ prices an Apex Pack at 100 Apex Coins, and EA's help page says packs are sold one at a time or 10 for 1,000 coins, with no discount for buying ten.

The price of the coins. On the PlayStation Store UK, 1,000 Apex Coins cost £8.99 and 11,500 cost £87.99. That puts one pack (100 coins) at:

• £8.99 ÷ 10 = £0.90 a pack if you buy coins 1,000 at a time
• £87.99 × 100 ÷ 11,500 = about 77p a pack at the largest coin bundle

Average packs to a Legendary. With the 30-pack guarantee, (1 − 0.926^30) ÷ 0.074 = 12.2 packs, which is about 1,217 coins.

Average cost of a Legendary item:

Buying coins as	Price per pack	Average cost (12.2 packs)	Most it can cost (30 packs)
1,000 coins for £8.99	£0.90	£10.94	£26.97
11,500 coins for £87.99	about 77p	£9.31	£22.95

Four things to keep in mind:

1. "A Legendary" means any Legendary item. The rate and the guarantee cover the rarity group. If you want one particular skin, the average cost is many times higher, and EA doesn't publish per-item odds for standard packs.
2. Coins come in fixed amounts. The cheaper rate in the second row means paying £87.99 up front. Coins you don't spend are still money spent.
3. The worst case is capped because of the guarantee. Without it, the average would be 13.5 packs and there would be no maximum at all.
4. Many packs are free. Apex Packs are also earned by levelling up and from the battle pass, so the table shows the cost only of packs you buy.

Where the sum gets harder

The sum needs both numbers from an official source, and they aren't always there.

The Elder Scrolls Online gets all the way, with a caveat. ZeniMax publishes a 2% chance of an Apex reward per Crown Crate, which is 50 crates on average, with no disclosed guarantee. Its Crown Crates announcement prices crates at 400 Crowns each or 5,000 for fifteen, and Crowns cost 0.52p to 0.77p each on the PlayStation Store UK. Fifty crates bought in 15-packs come to 16,667 Crowns, £87.29 to £128.67 on average for an Apex reward. The caveat is the date: the crate price comes from ZeniMax's 2016 launch announcement, and later season posts don't repeat it.

Black Desert gets most of the way but stops short of pounds. Pearl Abyss publishes a complete table for the Mystery Outfit Box: the top result, ten Premium Outfit Boxes, comes up 0.5% of the time, which is 200 boxes on average. Its Pearl Shop notice of 26 June 2024 sold two boxes for 2,600 Pearls, which is 1,300 Pearls a box, so 200 boxes come to 260,000 Pearls. That price is in PC Pearls, and the UK prices we can cite are for console Pearls, a separate currency, so the chain breaks at the last step.

Warframe is different again. Its Void Relics can't be bought, so the 2% rare chance of an Intact relic costs time rather than money: 50 relics on average.

When one of the two numbers is missing, the honest answer is the average number of openings and nothing more. A pound figure built on a price from a forum or a video is a guess.

A quick checklist before you buy

• Find the publisher's own rate for the exact box or banner, not a fan estimate.
• Check whether the rate is "at least one" per opening (as in Apex) or per item. "At least one" rates can't be added together across rarities.
• Find the guarantee, if any, and whether it carries over between events.
• Find the price of one opening in the game's currency and the price of that currency in pounds, at the pack size you would actually buy.
• Multiply. Then look at the worst case too. With no guarantee, there isn't one.

The drop-rate calculator works out the average number of openings and your chances at any count. Multiply the result by your price per opening.

How we checked

Rates and pack prices come from EA's and the other publishers' own pages, and coin prices from the PlayStation Store UK. All were read on 29 September 2026 and match our game pages. On 12 November 2026 we re-read EA's FAQ, ZeniMax's odds page and the PlayStation Store UK prices for Apex Coins and Crowns, and nothing had changed. Pearl Abyss's Black Desert pages wouldn't load that day, so those figures are as read on 29 September. We worked out every figure from those inputs and rounded only at the end.