Are the odds of winning a Premium Bonds prize puzzling when deciding where to put spare cash? Does the idea of a 50% or 90% chance feel abstract? This guide gives a direct, usable method to answer: how many Premium Bonds are needed to reach a chosen probability of winning over a single draw or across multiple monthly draws, with clear formulas, worked examples and strategy pointers to compare the result with a tax-free ISA (indicative figures at time of writing).
Key takeaways: what to know in one minute
- Core formula: probability of at least one win with B bonds in one draw = 1 − (1 − p)^B, where p = chance per £1 bond each draw. Use this to compute B for any target probability.
- Quick rule of thumb: for very small p, bonds required ≈ target constant / p (e.g. ~0.693/p for 50%).
- Monthly vs multi‑month: for M monthly draws, replace exponent B with B×M: 1 − (1 − p)^(B×M). Targets drop quickly as months increase.
- Expected return differs from win probability: expected average return equals the NS&I prize‑fund rate × invested amount, but distribution is skewed — many win nothing, few win big.
- Use an odds calculator: plug current NS&I odds (p) and desired probability (T) into the formula to get the bonds (B). Adjust if prize fund rate or odds change — figures here are indicative.
How many Premium Bonds to hit target odds: the maths and examples
The exact probability that at least one £1 bond wins a prize in a single monthly draw is p = 1 / X, where X is the published "1 in X" odds for a £1 bond. For B £1 bonds the chance of zero wins that month is (1 − p)^B. Therefore:
- probability of at least one win in one draw = 1 − (1 − p)^B
To find the number of bonds B needed to reach a target probability T (e.g. 0.5 for 50%) in one draw, rework the equation:
- B = ceil( ln(1 − T) / ln(1 − p) )
When p is very small (as with Premium Bonds), ln(1 − p) ≈ −p, so B ≈ ceil( −ln(1 − T) / p ). That approximation gives quick intuition.
Worked numeric examples (indicative odds)
Use the formula with three illustrative odds (indicative at time of writing): 22,000 to 1, 32,000 to 1 and 48,500 to 1. These are example X values; check NS&I for current published odds.
| target probability |
bonds needed @ 1 in 22,000 |
bonds needed @ 1 in 32,000 |
bonds needed @ 1 in 48,500 |
| 50% |
15,256 (£15,256) |
22,165 (£22,165) |
33,588 (£33,588) |
| 75% |
38,533 (£38,533) |
55,963 (£55,963) |
84,780 (£84,780) |
| 90% |
50,572 (£50,572) |
73,770 (£73,770) |
111,934 (£111,934) |
| 95% |
68,771 (£68,771) |
100,516 (£100,516) |
152,523 (£152,523) |
Notes: numbers are rounded up to whole bonds; 1 bond = £1. These examples are indicative and depend entirely on the current per‑bond odds X published by NS&I.
A simple calculator needs three inputs: current odds per £1 (X), desired probability (T as percent or decimal), and timeframe (single draw or number of monthly draws M). Steps:
- Enter X (e.g. 24,500 → p = 1/X).
- Choose T (e.g. 0.50 for 50%).
- Choose M (number of monthly draws you care about; for one month set M = 1).
- Compute B = ceil( ln(1 − T) / (M × ln(1 − p)) ).
If using the small‑p approximation: B ≈ ceil( −ln(1 − T) / (M × p) ). That is faster and accurate when p ≪ 1.
Example: target 90% over 12 months
- Suppose X = 24,500 so p = 1/24,500 ≈ 0.000040816.
- Target T = 0.90, M = 12.
- Exact: B = ceil( ln(0.10) / (12 × ln(1 − p)) ) ≈ ceil( 2.3026 / (12 × 0.000040816) ) ≈ ceil(2.3026 / 0.0004898) ≈ ceil(4,700) → 4,700 bonds (£4,700).
This shows the power of time: reaching 90% over 12 months may need a fraction of bonds compared with a single draw.
Expected return and win probability for Bonds: separating average yield from chance of prizes
Premium Bonds present two related but distinct metrics:
- Expected return (mean): NS&I publishes a prize‑fund rate which approximates the average annual return across all bondholders. For example, a prize fund of 3.0% means the expected average return is ~3.0% AER (indicative). This equals total prize value paid / total invested.
- Win probability per bond: the monthly per‑bond win probability p determines how likely an individual bond is to be among the winners in any draw. The distribution of prizes is skewed: most bonds win nothing each year, a small fraction win multiple prizes or larger prizes.
Practical implication: holding the number of bonds calculated to reach probability T does not guarantee a particular money outcome. It guarantees only the probability of receiving at least one prize in the period considered. Expected monetary return remains approximately the prize‑fund rate × invested amount (minus the effect of rounding and tax irrelevance since prizes are tax‑free), irrespective of how many bonds one holds.
Converting probability targets into expected money outcomes
If the goal is a cash expectation (average pounds per year), compare:
- Expected annual payout from bonds = prize fund rate × invested amount (indicative)
- Guaranteed interest from a cash ISA = ISA rate × invested amount (tax‑free similarly)
For example, with £10,000 invested and a prize fund rate of 3.0% the expected annual return ≈ £300. But with a target probability approach (say aiming for 50% monthly chance), the probability does not translate into a guaranteed cash amount — it describes chance of winning a prize, not expected pounds won.
When the aim is a steady expected income, a cash ISA at a competitive AER may be easier to model and predict.
Weighing tax‑free ISAs versus Premium Bonds when focusing on target odds
For a reader deciding between ISAs and Premium Bonds specifically to hit a probability target, consider three practical comparisons:
- Certainty vs chance: Cash ISAs provide a predictable AER and therefore predictable expected income. Premium Bonds provide probabilistic outcomes and tax‑free prizes but no guaranteed interest.
- Expected annual yield: compare the prize‑fund rate with the cash ISA AER (both tax‑free if ISA). If the ISA AER is higher than the prize fund rate, a cash ISA typically offers a better expected return with less variance.
- Psychological value and liquidity: Premium Bonds allow prize chasing and are instantly cashable via NS&I, but large prizes are rare; ISAs can offer better long‑term compounding for savers seeking steady growth.
Link to official ISA rules for confirmation: gov.uk: individual savings accounts.
How the NS&I prize fund rate changes outcomes: sensitivity and recalculation method
The prize fund rate determines the average monetary return but may be linked to the total number and size of prizes. If NS&I adjusts the prize fund rate, two primary effects occur:
- Expected return shifts proportionally. If the prize fund rate rises from 3.0% to 3.5%, the average pounds returned per £1 invested increases by ~0.5 percentage points.
- Odds or prize distribution may change. NS&I may keep prize categories fixed but adjust counts or sizes; this can change the effective p or the expected prize size per win.
Practical recalculation: when NS&I publishes new odds X' or a new prize fund rate, recompute p = 1 / X' and plug into earlier formulas for B. If only the prize fund rate changes but X stays constant, the bond counts for probability targets remain the same; the expected money outcome improves.
Always treat current odds and prize fund rates as indicative at time of writing and verify on the NS&I site: NS&I premium bonds.
How to convert a probability target into bonds
Step 1 ➜ find current odds per £1 (X)
Step 2 ➜ convert to p = 1 / X
Step 3 ➜ choose target probability (T) and months (M)
Step 4 ➜ compute B = ceil( ln(1 − T) / (M × ln(1 − p)) )
Result ➜ buy B bonds (1 bond = £1); track changes to X or prize fund rate and recalc
Step‑by‑step: build a bond strategy to meet goals (practical worksheet)
- Define the goal in probability terms: choose T (e.g. 50%, 75%, 90%).
- Choose timeframe: single monthly draw or M months (longer M reduces bonds needed).
- Check current odds X on NS&I and compute p = 1 / X. Use the live page: NS&I premium bonds.
- Calculate B using the formula and round up to a whole number of bonds.
- Translate B into cash = B × £1. Compare that cash with alternatives: a cash ISA at a quoted AER (see gov.uk ISA) to compare expected return.
- Decide on diversification: keep part in ISA for predictable returns and part in Premium Bonds if prize chance is desired.
Example strategy templates
- Conservative savers seeking steady expected income: prefer ISA when the ISA AER ≥ prize‑fund rate.
- Mixed strategy for entertainment + upside: hold an emergency buffer in ISA and use a capped amount in Premium Bonds sized to reach a satisfying probability target (e.g. a 50% monthly chance) for the remaining amount.
Advantages, risks and common mistakes
✅ benefits / when to apply
- Tax‑free prizes: winnings are tax‑free, which can be attractive for non‑ISA investors.
- Simplicity and liquidity: bonds are easy to buy, transfer and cash in via NS&I.
- Psychological value: many value chance of large, tax‑free prizes despite lower expected returns.
⚠️ errors to avoid / risks
- Confusing probability with guaranteed cash: hitting a 50% chance does not mean £X per year — it means a 50% chance of at least one prize in the chosen draw(s).
- Ignoring prize fund changes: prize fund and published odds change; recalculate after updates.
- Over‑concentration: buying extremely large numbers of bonds to chase very high probabilities may be an inefficient use of capital compared with interest‑bearing products.
Frequently asked questions
How many bonds for a 50% chance in one draw?
Use B = ceil( ln(0.5) / ln(1 − p) ). For example, if odds are 24,500 to 1, roughly 17,000 bonds (£17,000) are needed (indicative).
Yes. For M months, use B = ceil( ln(1 − T) / (M × ln(1 − p)) ). Months reduce required bonds because each draw is an independent chance.
Do prize fund rate changes alter the number of bonds required?
Only if NS&I changes the published per‑bond odds (X). If only the prize fund rate changes but X remains, the bond counts for probability targets stay the same; monetary expectation changes.
Is the expected return from Premium Bonds equal to the prize fund rate?
Yes, on average across all holders the expected monetary return equals the prize fund rate × invested amount. Individual outcomes vary widely.
Should savers use ISAs instead for steady returns?
If the priority is predictable, steady returns, a cash ISA with a competitive AER is often preferable. Use the probability method here only when the goal is chance‑based outcomes.
Where to verify official odds and prize fund rate?
Check NS&I for live odds and prize fund announcements: NS&I premium bonds.
Can children hold Premium Bonds and count for target odds?
Yes. Bonds held in a child’s name count the same; the formula applies identically. Consider the child ISA as an alternative for tax‑efficient saving.
Your next step:
- Identify the probability target (T) and timeframe (M) for the outcome that matters.
- Retrieve the current NS&I odds (X) and compute p = 1 / X; use B = ceil( ln(1 − T) / (M × ln(1 − p)) ).
- Compare the cash required (B × £1) with expected returns from a cash ISA and decide the split between bonds and ISA based on risk appetite.
Written by Alan White, UK‑based personal finance researcher. For official odds and prize fund details consult NS&I and for ISA rules consult gov.uk. This guide uses indicative figures; always confirm live rates before buying.