So we have an absorbed dose of 300mGy ( or 0.3 grays), this represents energy absorbed by the material divided by it's mass. But does not take into account the type of radiation or the tissue type. Effective dose does. 
We can use two formula to gain an effective dose for strontium 89 
Equivalent dose = radiation weighting factor x  absorbed dose
Effective dose = tissue weighting factor x equivalent dose 

So radiation weighing factors for alpha = 20, beta = 1 (these are known values, meaning alpha is 20x more harmful!) 

All we need to know to calculate our effective dose for strontium is the tissues weighting factor which we can calculate from the alpha result, as it's the same for all radiation amounts/types. 
We know an effective dose is 300mSv for an alpha absorbed dose of 300mGy in this tissue . 
So lets gain the Alpha equivalent dose and then the Tissue factor. 

Alpha equivalent dose =  20 x 300mGy = 6000mSv
Tissue factor = 300mSv / 6000mSv = 0.05

So Beta equivalent dose = 1 x 300mGy = 300mSv
Beta effective dose = 0.05 x 300mSv = 15mSv

So our answer is 0.015 Sieverts. (or 15 Milisieverts) 
(I.e 20 times less than the alpha amount due the radiation weighting factor)

English

A patient receives absorbed dose of radiation 300mGy from an alpha emitting radionuclide, Radium 223, into breast tissue this emits 300mSv. The patient at the same time receives a dose (300mGy) from strontium 89 of a beta emitter calc effective dose

See my answer from same previously submitted question

A patient receives absorbed dose of radiation 300mGy from an alpha emitting radionuclide, radium 223, into breast tissue this emits 300mSv. The patient at the same time receives a dose (300mGy) from strontium 89 of a beta emitter. Calm effective dose

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