The purpose of this series is developing understanding of stastical problems in AS and A level maths. Let's get started with a simple concept in statistics: Mutually exclusive events.

The probability of an OR event is calculated by the following rule:

`P(A || B) = P(A) + P(B) - P(A && B)`

The probability of event A or event B happening is equal to the probability of event A plus the probability of event B minus the probability of event A and event B happening simultaneously.

Mutually exclusive events are events that cannot happen at the same time. For example, the head and tail results of a toin coss are mutually exclusive because they can't both happen at once. Thus, the above example for a coin toss would look like this:

`P(H || T) = P(H) + P(T) - P(H && T)`

Note that the probaility of tossing a coin and the result being both head and tails is 0.

`P(H || T) = (0.5) + (0.5) - (0)`
`P(H || T) = 1`

Thus the probability of a coin toss result being a heads or a tails is 1, in other words: certain.


Your task:

You are going to have to work out the probability of one roll of a die returning two given outcomes, or rolls. Given that dice rolls are mutually exclusive, you will have to implement the above forumala. To make this interesting (this is a coding challenge after all), these dice are not fair and thus the probabilites of receiving each roll is different.

You will be given a two-dimensional array containing the number each of the results (1-6) of the die and the probability of that roll for example `[1 , 0.23]` as well as the two rolls for example `1` and `5`. 

Given the two roll probabilities to calculate, return the probability of a single roll of the die returning either. If the total probability of the six rolls doesn't add up to one, there is a problem with the die; in this case, return null. 
Return your result as a string to two decimal places.

Example below:

`1 : 1/6`

`2 : 1/6`

`3 : 1/6`

`4 : 1/6`

`5 : 1/6`

`6 : 1/6`

If asked for the rolls `1` and `2` then you would need to sum the probabilities, both `1/6` therefore `2/6` and return this. As above, you will need to return it as a decimal and not a  fraction.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "def mutually_exclusive(dice, call1, call2):\n    if sum(prob for _, prob in dice) != 1:\n        return None\n    x = dict(dice)\n    return format(x[call1] + x[call2], '.2f')"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'mutually_exclusive'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 5560