While chance plays a role in determining whether you win big on the Premium Bonds, basic probability means that the more you hold of these investments, the more likely and frequently you will win a prize.

The latest move to encourage savers to put their money into these bonds, offered by National Savings and Investments, is increasing the maximum investment limit from £40,000 to £50,000 on 1 June 2015.

This follows the changes made last year- doubling the jackpot £1m prize on offer each month and raising the limit on the amount an individual could invest from £30,000 to £40,000.

Premium Bonds allow you to save from £100 with easy access to your cash. However, they aren’t like normal savings accounts as they don’t pay a regular return on your money.

Instead the interest that should be paid is used to fund a monthly prize draw, where bond holders can win between £25 and £1m or, of course, nothing at all.

It is worth noting that regardless of the increase in the maximum investment limit, the prize fund interest rate is not changing from its current level of 1.35 per cent. Before I go any further, let me put that into context.

At 1.35 per cent, the minimum investment of £100 each month would theoretically generate just over 11 pence in interest, which would be added to the prize fund. Given that the minimum prize for each draw is £25, then in simple terms, a bond holder with £100 invested could expect to wait 222 months, that’s over 18 years, before ‘average luck’ dictates that they should win a prize.

This sounds like a particularly long time to get a return, but remember that each and every bond has the same chance of winning the jackpot every month. Precisely, this is 26,000 to 1 for each £1 bond number.

Clearly, the more bonds held, the greater the chances of winning any prize. Using the same simple theory as above, if our bond holder had invested £1,000, then average luck dictates they would only have to wait 22 months to win any prize. However, it must be appreciated that this still translates to just 1.35 per cent.

For people who already hold the maximum in Premium Bonds, have further capital to invest and won prizes, then putting an extra £10,000 in these investments when the maximum increases will probably be on their agenda.

Should you buy the maximum of £50,000 in Premium Bonds?

If you are comfortably affluent and alongside other investments, you have large cash holdings and broadly diversified portfolios, Premium Bonds are likely to be part of your holdings. As such, increasing your holdings to the £50,000 maximum is likely to be up for careful consideration. After all, compared to a deposit account earning 1 per cent gross interest, the ‘opportunity cost’ to a higher rate taxpayer of holding a further £10,000 of bonds is just £60 per annum – which with ‘average luck’ should translate to tax-free annual winnings of £135.

If you’re not awash with spare funds, the decision becomes a little more specific to your circumstances and investment opportunities elsewhere. Although the ‘opportunity cost’ is currently very low, it is slightly higher for a basic rate taxpayer. If you pay tax at the basic rate, giving up 1 per cent gross interest on a £10,000 translates to an annual ‘loss’ of £80 per annum in return for the potential of winning up to £1,000,000 every month – but with a more likely average outcome of winning £125 (as winnings are in £25 denominations) per year.

Winnings from Premium Bonds are completely tax-free and another likely place for tax-free savings is investing in an ISA. However, you are currently limited to saving £15,240 into an Isa each tax year, while the limit on Premium Bonds is much higher. However, the benefits of saving within an ISA should not be overlooked, especially given all the recent changes to the scheme, and savers are likely to use both Premium Bonds and ISAs.

Another advantage to saving with Premium Bonds is that 100 per cent of your money is protected by HM Treasury. So while no investment is completely risk-free, these bonds are probably the closest you are going to get.

Also, despite Consumer Prices Index inflation rate remaining at 0 per cent, while the Retail Prices Index crept down from 1 per cent in February to 0.9 per cent in March, inflation risk over the long-term should not be ignored. By reference to the inflation rate using RPI, holders of Premium Bonds who have won nothing in the last five years have suffered erosion of the spending value of their capital over that same period to the tune of a 16.5 per cent.

No investment or savings solutions is perfect, but for many Premium Bonds are likely to be considered as part of their diversified portfolio, especially since receiving winnings is fun and the attractiveness of the bonds has increased with the changes to the limits and prizes.

For those who can’t invest the maximum in the bonds, the frequency of wins will be lower, but excluding very small holdings, it only takes a little bit of luck for such winnings to produce a better net return than most deposit accounts currently can.

However, a good investment manager will arrange a properly constructed portfolio of investments which is highly unlikely to earn you am, but is significantly more likely to return more than 1.35 per cent per annum over the longer term. However, you need to accept that ‘likely’ does not mean certainty and so luck plays a part there too. Talk to a good investment manager that has more luck and proven skill than others.